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+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nThe 20th century has seen the development of two pillars of modern theoretical physics: quantum field theory (QFT) and general relativity (GR). The standard model of particle physics, which successfully describes the quantum properties of the strong and electroweak interactions, is based on the former framework. However, its na\\\"{i}ve application to the latter yields a QFT of gravity prone to (perturbatively) non-renormalizable ultraviolet (UV) divergences~\\cite{tHooft:1974toh,Goroff:1985th,Goroff:1985sz}.\n\nDespite remarkable progress in a number of directions, the difficulties of formulating a complete theory of quantum gravity have led a considerable portion of the community to shift the focus on general, possibly model-independent lessons that could shed light on the nature of gravity at all scales. Many of these proposals, commonly dubbed ``swampland conjectures''~\\cite{Vafa:2005ui} in the context of string theory, rest on considerations on black-hole physics, which often can lie entirely in the semi-classical regime where one expects low-energy effective field theory (EFT) to be a reliable description.\n\nOn the other hand, some aspects of these conjectures arose from and are tied to string theory and, in particular, its spacetime-supersymmetric incarnations. These settings are much better understood, since quantum corrections are often under quantitative control~\\cite{Berglund:2005dm, Gonzalo:2018guu, Marchesano:2019ifh, Blumenhagen:2019vgj, Baume:2019sry, Palti:2020qlc, Marchesano:2021gyv, Lee:2018urn, Lee:2018spm, Lee:2019tst, Lee:2019xtm} and can sometimes even be computed exactly~\\cite{Klaewer:2020lfg, Klaewer:2021vkr}. In certain settings, such as $\\mathcal{N} = 2$ Calabi-Yau flux compactifications, one can even classify general families of models at once~\\cite{Grimm:2018ohb, Grimm:2019wtx, Grimm:2019ixq, Gendler:2020dfp, Grimm:2020ouv, Bastian:2021eom}. In some string models with high-energy supersymmetry breaking, a variety of swampland proposals were verified~\\cite{Basile:2020mpt, Basile:2021mkd}, although a construction of realistic and (meta-) stable vacua is still an open problem~\\cite{Kachru:2003aw, Balasubramanian:2005zx, Koerber:2007xk, Danielsson:2009ff, Moritz:2017xto, Kallosh:2018nrk, Bena:2018fqc, Gautason:2018gln, Cordova:2018dbb, Blaback:2019zig, Hamada:2019ack, Gautason:2019jwq, Cribiori:2019clo, Andriot:2019wrs, Shukla:2019dqd, Shukla:2019akv, Cordova:2019cvf, Andriot:2020wpp, Andriot:2020vlg,Farakos:2020wfc,Gao:2020xqh,Bena:2020qpa,Bena:2020xrh,Crino:2020qwk,Dine:2020vmr,Basiouris:2020jgp,Cribiori:2020use,Hebecker:2020ejb,Andriot:2021rdy,DeLuca:2021pej,Cicoli:2021dhg,Cribiori:2021djm}. At any rate, it is paramount to understand the consequences of general consistency conditions outside of the specific contexts arising from (supersymmetric) string compactifications, although parametric control is most likely going to be problematic~\\cite{Dine:1985he,Bena:2018fqc,Gao:2020xqh,Bena:2020xrh,Dine:2020vmr} due to unknown or uncalculable corrections. To wit, efforts to test swampland proposals have almost entirely focused on supersymmetric settings, and in particular on stringy constructions. In order to shed light on whether they only encode a ``string lamppost principle''~\\cite{Montero:2020icj} or they hold in more generality, it is important to extend their exploration to a broader class of quantum gravity models. On the other hand, the more stringent and well-grounded swampland proposals, such as the ``no global symmetries''~\\cite{Misner:1957mt, Polchinski:2003bq, Banks:2010zn, Harlow:2018tng} and weak gravity~\\cite{Arkani-Hamed:2006emk} conjectures, could help guide the search for asymptotic safety, which is currently faced with the daunting prospect of navigating ever-larger theory spaces~\\cite{Benedetti:2010nr, Knorr:2021slg}.\n\nA concrete problem that can be phenomenologically relevant in the near future is constraining the possible values of the Wilson coefficients of a curvature expansion of the gravitational EFT. In particular, one can expect that generic detectable leading-order effects of quantum gravity be encoded in the coefficients of the quadratic curvature invariants, which we shall discuss in detail in the following, or in some specific non-local form factors~\\cite{Belgacem:2017cqo,Knorr:2018kog}. Some efforts in this direction have been made using the S-matrix bootstrap~\\cite{Guerrieri:2021ivu, Caron-Huot:2021rmr}, finding compelling agreement with the parameter space allowed by string theory. In the EFT framework, the problem has also been investigated via positivity bounds~\\cite{deRham:2017zjm, deRham:2017xox, deRham:2018qqo, DeRham:2018bgz, Alberte:2019lnd, Alberte:2019xfh, Alberte:2019zhd, Alberte:2020bdz, Alberte:2020jsk, deRham:2021fpu}.\n\nIn this paper we approach this issue from a novel direction, studying the constraints coming from swampland conjectures together with the consistency conditions required by the existence of a UV fixed point of the gravitational renormalization group (RG) flow\\footnote{See also~\\cite{deAlwis:2019aud} for related discussions on the weak gravity conjecture in the context of asymptotically safe gravity.}. The latter scenario has been termed ``asymptotic safety'', in analogy with asymptotic freedom as a particular case. This idea, originally due to Weinberg~\\cite{1976W}, would imply that the Wilson coefficients of the IR effective action stem from a UV-complete RG trajectory, which in turn would be determined by a finite number of relevant deformations from the fixed point. Recently, this area of research has witnessed considerable development of the theoretical framework to investigate RG flows beyond perturbation theory~\\cite{Dupuis:2020fhh} and finding evidence for the existence of the Reuter fixed point in a variety of different approximation schemes~\\cite{Souma:1999at,Lauscher:2002sq,Litim:2003vp,Codello:2006in,Machado:2007ea,Benedetti:2009rx,Dietz:2012ic,Dona:2013qba,Eichhorn:2013xr,Dona:2014pla,Christiansen:2014raa,Falls:2014tra,Christiansen:2015rva,Meibohm:2015twa,Oda:2015sma,Dona:2015tnf,Biemans:2016rvp,Eichhorn:2016esv, Dietz:2016gzg, Falls:2016msz,Gies:2016con,Biemans:2017zca,Christiansen:2017cxa,Hamada:2017rvn,Platania:2017djo,Falls:2017lst,deBrito:2018jxt,Eichhorn:2018nda,Eichhorn:2019yzm,deBrito:2019umw,Knorr:2021slg} (see also~\\cite{Donoghue:2019clr,Bonanno:2020bil} for critical assessments on the status of the field and its open questions). Possible implications of asymptotically safe gravity in astrophysics and cosmology (see~\\cite{Bonanno:2017pkg,Platania:2020lqb} for reviews) have been investigated using simplified models~\\cite{Bonanno:2006eu,Falls:2012nd,torres15,Koch:2015nva,Bonanno:2015fga,Bonanno:2016rpx,Kofinas:2016lcz,Falls:2016wsa,Bonanno:2016dyv,Bonanno:2017gji,Bonanno:2017kta,Bonanno:2017zen,Bonanno:2018gck,Liu:2018hno,Majhi:2018uao,Anagnostopoulos:2018jdq,Adeifeoba:2018ydh,Pawlowski:2018swz,Gubitosi:2018gsl,Platania:2019qvo,Platania:2019kyx,Bonanno:2019ilz,Held:2019xde} and more elaborate computations~\\cite{Bosma:2019aiu}, leading to the tentative conclusions that black-hole and cosmological singularities could be resolved by quantum effects, and that the nearly scale-invariant  cosmological power spectrum could arise naturally from a nearly scale-invariant  asymptotically safe regime.\n\nIn this paper we shall propose a concrete method to extract the allowed region of IR parameters from the RG flow of asymptotically safe trajectories. \nIn particular, we shall focus on the simpler case of the one-loop approximation in quadratic gravity~\\cite{Codello:2006in,Niedermaier:2009zz, Niedermaier:2010zz} in order to test our construction and provide a proof of principle of our idea. We will show that the IR limit of asymptotically safe trajectories falls inside the region allowed by the weak gravity conjecture and electromagnetic duality, and display a non-trivial intersection with the one allowed by the de Sitter and trans-Planckian censorship bounds.\n\nThe contents of this paper are organized as follows. In sect.~\\ref{sec:swampland} we provide a brief overview of swampland conjectures, focusing on the weak gravity conjecture, the de Sitter conjecture and the trans-Planckian censorship conjecture, since they entail the most relevant bounds for our subsequent analysis. In sect.~\\ref{sec:one-loop} we describe in detail the one-loop approxima\\-tion to quadratic gravity that we employ as testing grounds, and our method of extracting the physical IR Wilson coefficients. The resulting effective action turns out to contain non-local form factors. In sect.~\\ref{sec:results} we collect our results: in sect.~\\ref{sec:IR_space} we present the allowed region of parameter space that we found, which spans a plane in the three-dimensional space of dimensionless IR parameters, and in sect.~\\ref{sec:wgc_constraints} and sect.~\\ref{sec:dsc_constraints} we study the constraints stemming from the swampland conjectures discussed in sect.~\\ref{sec:wgc_intro} and sect.~\\ref{sec:dsc_tcc_intro} respectively. In sect.~\\ref{sec:intersections} we discuss and display the intersection of all regions. We conclude with a summary and some perspectives in sect.~\\ref{sec:conclusions}.\n\n\\section{An overview of swampland conjectures}\\label{sec:swampland}\n\nAs we have anticipated in the introduction, swampland conjectures are proposals that ought to rule out EFTs of gravity that do not admit UV completions~\\cite{Vafa:2005ui}. These conjectures are generally motivated in part by purely low-energy considerations, stemming from black-hole physics or inflation, but they also arise from detailed investigations of string-theoretic settings, where generally one has more control over corrections and patterns can be corrobo\\-rated across families of EFTs. The latter approach has led some to describe a ``lamppost'' effect~\\cite{Montero:2020icj}, whereby only settings that are somewhat under control can be investigated and thus it is unclear to which extent the resulting conclusions can be generalized. Furthermore, while at least minimal supersymmetry is generally unbroken in order to retain computational control, recent considerations~\\cite{Cribiori:2021gbf,Castellano:2021yye} point to a tension between low-energy supersymme\\-try breaking\\footnote{Nevertheless, scenarios with high-energy supersymmetry breaking have been investigated in the context of the swampland~\\cite{Bonnefoy:2018tcp,Basile:2020mpt,Basile:2021mkd}. See~\\cite{Mourad:2017rrl, Basile:2021vxh, Mourad:2021lma} for recent reviews.} and the consistency of the EFT. As we have discussed in the preceding section, one of the motivations behind this work is indeed to go beyond the usual settings, seeking lessons for other approaches to quantum gravity.\n\nSince its inception, the swampland program aims to describe the boundary between the landscape of consistent gravitational EFTs with a growing number of proposed criteria\\footnote{See~\\cite{Palti:2019pca, vanBeest:2021lhn} for reviews.}, numerous relations among which~\\cite{Andriot:2020lea, Lanza:2020qmt} point to a deeper underlying principle. In particular, connections between the distance conjecture~\\cite{Ooguri:2006in, Ooguri:2018wrx} and string dualities suggest that an organizing principle for these consistency criteria in the IR be related to a non-perturbative UV formulation of quantum gravity. Furthermore, as we shall see in the following, swampland considerations have provided intriguing clues toward a number of phenomenological puzzles~\\cite{Grana:2021zvf}. \n\nIn this paper we shall focus on some conjectures which can provide bounds for the Wilson coefficients of the gravitational EFT. In particular,\n\n\\begin{itemize}\n    \\item The \\emph{weak gravity conjecture} (WGC)~\\cite{Arkani-Hamed:2006emk} relates the mass and charge of light states and black holes;\n    \\item The \\emph{de Sitter conjecture} (dSC)~\\cite{Obied:2018sgi}, along with its refined versions~\\cite{Ooguri:2018wrx,Garg:2018reu,Andriot:2018mav}, constrains the behavior of scalar potentials and their derivatives, leading to an obstruction to the existence of de Sitter vacua that is $\\mathcal{O}(1)$ in Planck units;\n    \\item The \\emph{trans-Planckian censorship conjecture} (TCC)~\\cite{Bedroya:2019snp, Brandenberger:2021pzy} constrains sub-Planckian cosmological perturbations to remain sub-Planckian across inflation, and leads to bounds on the lifetime of metastable de Sitter configurations as well as on the $\\mathcal{O}(1)$ parameter that appears in the dSC, at least in asymptotic regions of field space.\n\\end{itemize}\n\nIn light of the latter consideration, for the purposes of this paper in the following we shall investigate the consequences of the TCC on Starobinsky-like inflationary potentials as a special case of the dSC. Indeed, we shall restrict ourselves to the asymptotic region of field space corresponding to small curvatures in Planck units, where the TCC could provide a dSC bound with a specific $\\mathcal{O}(1)$, as we shall see below.\n\n\\subsection{Weak gravity conjecture and black holes}\\label{sec:wgc_intro}\n\nLet us begin reviewing some features of the (electric) WGC, referring the reader to~\\cite{Palti:2019pca,  vanBeest:2021lhn} for more details. In its most basic form, it states that in a consistent EFT of gravity coupled to a $U(1)$ gauge field there exists a state whose mass $m$ is lower than its charge $q$ in Planck units. In four dimensions, the bound for charged particles reads\n\\begin{eqaed}\\label{eq:wgc_basic}\n    \\frac{m}{M_\\text{Pl}} \\leq \\mathcal{O}(1) \\, q \\, ,\n\\end{eqaed}\nwhere the model-dependent $\\mathcal{O}(1)$ constant is $\\frac{1}{\\sqrt{2}}$ in Einstein-Maxwell theory.\n\nAmong various motivations and evidence gathered in the literature, the WGC is grounded in black-hole physics from the requirement that charged, extremal black holes be able to decay, lest protected by a symmetry (such as supersymmetry, in the case of BPS-saturated states). The rationale behind this lies in avoiding remnants while keeping the black hole from violating the extremality bound, since a violation of either would presumably lead to consistency issues potentially within the EFT regime~\\cite{Giddings:1992hh, Susskind:1995da, Arkani-Hamed:2006emk}. For charged black holes of mass $M$ and charge $Q$, this requirement translates into\n\\begin{eqaed}\\label{eq:wgc_bhs}\n    \\frac{M}{Q} \\geq \\left(\\frac{M}{Q}\\right)_\\text{extremal} \\, ,\n\\end{eqaed}\nwhere the latter is generally an $\\mathcal{O}(1)$ constant\\footnote{For Einstein-Maxwell theory, the extremality bound reads $\\frac{M}{Q} \\geq \\sqrt{2} \\, M_\\text{Pl}$.}. However, higher-curvature corrections could potentially spoil this condition even for macroscopic black holes, provided they are sufficiently close to extremality. Writing the leading quartic corrections according to~\\cite{Kats:2006xp}\n\\begin{eqaed}\\label{eq:eft_corr}\n    \\Delta \\mathcal{L} = c_1 \\, R^2 + c_2 \\, R_{\\mu \\nu}R^{\\mu \\nu} + c_3 \\, R_{\\mu \\nu \\rho \\sigma} R^{\\mu \\nu \\rho \\sigma} \\, ,\n\\end{eqaed}\nthe resulting bounds for the corresponding Wilson coefficients $c_i$ comprise a family of inequalities for linear combinations of the $c_i$, parametrized by the extremality parameter of the black hole~\\cite{Arkani-Hamed:2006emk,Kats:2006xp,Cheung:2018cwt,Charles:2017dbr,Hamada:2018dde,Charles:2019qqt}. The extremality bound in general now takes the form\n\\begin{eqaed}\\label{eq:wgc_bhs_hd}\n    \\frac{M}{Q} \\geq \\left(\\frac{M}{Q}\\right)_\\text{extremal} \\left(1 - \\frac{\\Delta}{M^2} \\right) \\, ,\n\\end{eqaed}\nwhere the linear combination $\\Delta$ of Wilson coefficients is to be non-negative in order for the WGC to hold, and is proportional to the coefficient $c_2 + 4c_3$ of the Weyl-squared term~\\cite{Kats:2006xp,Charles:2019qqt}.\n\nThe leading order contributions to $\\Delta$ comprise not only the Wilson coefficients in the effective action of eq.~\\eqref{eq:eft_corr}, but also the Wilson coefficients that involve the $U(1)$ gauge field. It has been recently shown~\\cite{Cano:2021tfs} that, \\emph{assuming invariance under electromagnetic duality}, higher-curvature corrections up to sextic order can be written in terms of purely gravitational terms, up to field redefinitions. Let us stress that our aim is to intersect swampland bounds with the constraints provided by asymptotic safety, and the technical obstacles to compute its consequences for quartic electromagnetic couplings in gravity, which would entail involved FRG computations along the lines of~\\cite{Knorr:2021slg}, compel us to focus on the duality-invariant scenario of~\\cite{Cano:2021tfs}, which at any rate appears intriguing on its own\\footnote{Another instance of the interplay between duality and the WGC has been studied in~\\cite{Loges:2020trf}.}. Moreover, the electromagnetic couplings do not run under the RG flow at one loop because of tree-level duality~\\cite{Charles:2017dbr,Charles:2019qqt}. This has been used to argue that the low-energy behavior of the correction $\\Delta$ to the extremality bound is dominated by the Weyl anomaly coefficients that drive the running of the $c_i$, and in particular of the Weyl-squared Wilson coefficient $c_2 + 4c_3$, as we shall see in the following. However, in the present case we would like to constrain the physical parameters built out of the Wilson coefficients and the Planck scale, in order to compare the resulting bounds with the constraints of asymptotic safety. We shall describe the procedure in detail in the following section.\n\n\\subsection{de Sitter and trans-Planckian censorship}\\label{sec:dsc_tcc_intro}\n\nLet us now move on to discuss the dSC and the TCC. The former quantifies an obstruction to the existence of de Sitter vacua, in the form of a bound for the (field-space gradient of the) scalar potential $V(\\phi)$. Indeed, since in this setting de Sitter vacua would arise as positive-energy critical points of $V$, a natural bound that would prevent these takes the form\n\\begin{eqaed}\\label{eq:dsc_basic}\n    M_\\text{Pl}\\, \\abs{\\nabla V} \\geq c V \n\\end{eqaed}\nfor field ranges\n\\begin{eqaed}\\label{eq:field_range}\n    \\Delta \\phi \\lesssim f \\, M_\\text{Pl} \\, ,\n\\end{eqaed}\nwhere $c \\, , \\, f > 0$ are (\\emph{a priori} model-dependent) constants. Within the EFT framework, their natural values are $\\mathcal{O}(1)$, indicating that the obstruction is tied to the expected cutoff of the EFT. However, one is readily confronted with a tension between the bound in eq.~\\eqref{eq:dsc_basic} and slow-roll inflation~\\cite{Garg:2018reu, Kinney:2018nny}\\footnote{See also~\\cite{Rudelius:2019cfh,Rudelius:2021oaz,Chojnacki:2021fag,Jonas:2021xkx} for discussions on eternal inflation and the swampland.}, leading to refinements involving the Hessian matrix of the potential~\\cite{Garg:2018reu,Ooguri:2018wrx}. In particular, whenever the bound of eq.~\\eqref{eq:dsc_basic} would be violated, the matrix\n\\begin{eqaed}\\label{eq:ref_dsc}\n    M_\\text{Pl}^2\\, \\text{Hess}(V) + \\, c' \\, V\n\\end{eqaed}\nwould be negative semidefinite, with $c' > 0$ another $\\mathcal{O}(1)$ constant. Further refinements were proposed in~\\cite{Andriot:2018mav}, but in our setting we shall find that the first bound of eq.~\\eqref{eq:dsc_basic} is sufficient, since it encompasses eq.~\\eqref{eq:ref_dsc} in the regions of parameter space that we are concerned with.\n\nOn the other hand, the TCC surmises that sub-Planckian quantum fluctuations in the early universe at initial time $t_i$ never grow macroscopic at a final time $t_f$. In particular, they ought to never cross the Hubble horizon and freeze. This requirement can be formulated, in terms of the scale factor $a(t)$ and the corresponding Hubble parameter $H$, by~\\cite{Bedroya:2019snp, Brandenberger:2021pzy}\n\\begin{eqaed}\\label{eq:tcc_basic}\n    \\frac{a(t_f)}{a(t_i)} \\lesssim \\frac{M_\\text{Pl}}{H(t_f)} \\, ,\n\\end{eqaed}\nagain up to an $\\mathcal{O}(1)$ constant. An intriguing consequence of eq.~\\eqref{eq:tcc_basic} is that de Sitter configurations are not prohibited, but they are metastable with a lifetime $T$ bounded by\n\\begin{eqaed}\\label{eq:tcc_life}\n    T \\lesssim \\frac{1}{H} \\, \\log \\frac{M_\\text{Pl}}{H} \\, ,\n\\end{eqaed}\nof the order of a trillion years. This results points to a possible resolution of the coincidence problem in this setting.\n\nThe most relevant consequence of the TCC for the purposes of this paper is that, in the presence of a scalar potential, it leads to a bound of the form of eq.~\\eqref{eq:dsc_basic} with\n\\begin{eqaed}\\label{eq:tcc_c1}\n    c = \\frac{2}{\\sqrt{(d-1)(d-2)}}\n\\end{eqaed}\nin $d$ spacetime dimensions, at least in asymptotic regimes of field space. In the present setting, the scalar potential arises from the quadratic curvature terms, and the corresponding asymptotic regime for gravitational field fluctuations is that of small curvatures~\\cite{Lust:2019zwm}. This regime is mapped to a neighbourhood of the origin in the inflaton description. For generic curvatures, one expects that both the purely gravitational description and the inflaton description be modified, including the geometry of field space. Nevertheless, since our current setup does not allow for precise quantitative bounds, we shall henceforth take eq.~\\eqref{eq:tcc_c1} simply as a reference point around which to study the more general bound of eq.~\\eqref{eq:dsc_basic}. Let us also remark that this value appears in a number of related swampland bounds~\\cite{Andriot:2020lea} and is well-behaved under dimensional reduction~\\cite{Rudelius:2021oaz}, and thus it may play a more prominent role in the story. At any rate, it would be interesting to explore the more direct implications of eq.~\\eqref{eq:tcc_basic} studying cosmological solutions or exploring the considerations of~\\cite{Bedroya:2019tba, Brandenberger:2021pzy} within our setup.\n\n\\section{One-loop RG flow in quadratic gravity}\\label{sec:one-loop}\n\nLet us now discuss the concrete setting in which we shall compute the possible values of the Wilson coefficients of the effective gravitational action. In this work we focus on the quadratic truncation\\footnote{Let us remark that here ``quadratic'' refers to the order in the curvatures. In terms of derivatives, the action in eq.~\\eqref{eq:quadratic_lagrangian} is quartic.}, in the one-loop approximation. In Euclidean signature, the Lagrangian pertaining to the full quadratic truncation reads\n\\begin{eqaed}\\label{eq:quadratic_lagrangian}\n    \\mathcal{L}=\\frac{2\\Lambda-R}{16\\pi G}+\\frac{1}{2\\lambda}\\,C^2-\\frac{\\omega}{3\\lambda}R^2+\\frac{\\theta}{\\lambda}E \\, ,\n\\end{eqaed}\nwhere $C^2 \\equiv C_{\\mu\\nu\\rho\\sigma}C^{\\mu\\nu\\rho\\sigma}$ is the square of the Weyl tensor, $E$ is the Gauss-Bonnet density and and the Wilson coefficients\n\\begin{eqaed}\\label{eq:wilson_coeffs}\n    g_C \\equiv \\frac{1}{2\\lambda} \\, , \\qquad\n    g_R \\equiv - \\, \\frac{\\omega}{3\\lambda}\n\\end{eqaed}\ncan be related to the $c_i$ coefficients in eq.~\\eqref{eq:eft_corr}. Indeed, since\n\\begin{eqaed}\\label{eq:weyl_gb}\n    C^2&=R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}-2R_{\\mu\\nu}R^{\\mu\\nu}+\\frac{R^2}{3}\\,,\\\\\n    E&=R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}-4R_{\\mu\\nu}R^{\\mu\\nu}+R^2\\,,\n\\end{eqaed}\nthe $c_i$ are related to the couplings in eq.~\\eqref{eq:quadratic_lagrangian} according to\n\\begin{eqaed}\\label{eq:couplings_relation}\n    c_1&=\\frac{1}{6\\lambda}-\\frac{\\omega}{3\\lambda}+\\frac{\\theta}{\\lambda} \\, ,\\\\\n    c_2&=-\\frac{1}{\\lambda}-\\frac{4\\theta}{\\lambda} \\,,\\\\\n    c_3&=\\frac{1}{2\\lambda}+\\frac{\\theta}{\\lambda} \\,.\n\\end{eqaed}\nWhile this setup holds in general spacetime dimensions $d$, we now restrict to $d = 4$. The one-loop beta functions of the couplings of eq.~\\eqref{eq:quadratic_lagrangian} read~\\cite{Codello:2006in,Niedermaier:2009zz, Niedermaier:2010zz}\n\\begin{eqaed}\\label{eq:betas}\n\\beta_{\\widetilde{\\Lambda}} & =-\\,2 \\widetilde{\\Lambda}+\\frac{1}{(4 \\pi)^{2}}\\left[\\frac{1+20 \\omega^{2}}{256 \\pi \\widetilde{G} \\omega^{2}} \\lambda^{2}+\\frac{1+86 \\omega+40 \\omega^{2}}{12 \\omega} \\lambda \\widetilde{\\Lambda}\\right] \\\\\n& \\quad -\\frac{1+10 \\omega^{2}}{64 \\pi^{2} \\omega} \\lambda +\\frac{2 \\widetilde{G}}{\\pi}-\\frac{83+70 \\omega+8 \\omega^{2}}{18 \\pi} \\widetilde{G} \\widetilde{\\Lambda} \\, , \\\\\n\\beta_{\\widetilde{G}} & =2 \\widetilde{G}-\\frac{1}{(4 \\pi)^{2}} \\frac{3+26 \\omega-40 \\omega^{2}}{12 \\omega} \\lambda \\widetilde{G}-\\frac{83+70 \\omega+8 \\omega^{2}}{18 \\pi} \\widetilde{G}^{2} \\, ,\n\\\\\n\\beta_{\\lambda} & =-\\frac{1}{(4 \\pi)^{2}} \\frac{133}{10} \\lambda^{2} \\, , \\\\\n\\beta_{\\omega} & =-\\frac{1}{(4 \\pi)^{2}} \\frac{25+1098 \\,\\omega+200 \\,\\omega^{2}}{60} \\lambda \\, , \\\\\n\\beta_{\\theta} & =\\frac{1}{(4 \\pi)^{2}} \\frac{7\\,(56-171\\, \\theta)}{90} \\lambda\n\\end{eqaed}\nwhere $\\widetilde{G}_k=G_k\\,k^2$ and $\\widetilde{\\Lambda}_k=\\Lambda_k\\,k^{-2}$\nare the dimensionless Newton coupling and cosmological constant respectively, and we have suppressed the subscript $k$ in eq.~\\eqref{eq:betas} for the sake of clarity.\n\nIn our setting, the flow of the (classically) marginal couplings $\\lambda$, $\\omega$ and $\\theta$ is decoupled from that of the Einstein-Hilbert couplings. Out of the UV fixed points\n\\begin{eqaed}\\label{eq:fixed_points}\n    \\lambda_\\ast = 0 \\, , \\qquad \\omega_\\ast = \\omega_\\pm \\equiv \\frac{-549 \\pm 7\\sqrt{6049}}{200} \\, , \\qquad \\theta_\\ast = \\frac{56}{171} \\, ,\n\\end{eqaed}\nUV completeness selects $\\omega_\\ast = \\omega_+ \\approx - \\, 0.023$~\\cite{Codello:2008vh}, which the solutions approach as the RG time\\footnote{Note that, since we are interested in the IR regime, our convention for the RG time is such that $t \\to +\\infty$ in the IR.} $t \\equiv \\log\\frac{k_0}{k} \\to -\\infty$, as is apparent from fig.~\\ref{fig:flowlambdaomega}. Let us remark that this fixed point is asymptotically safe, \\emph{i.e.} at least one coupling is not asymptotically free~\\cite{Codello:2008vh, Niedermaier:2009zz, Niedermaier:2010zz, Groh:2011vn}. Indeed, the critical exponents of $G$ and $\\Lambda$ are 2 and 4\\footnote{Although 2 and 4 are not the canonical mass dimensions of $G$ and $\\Lambda$, they are the canonical dimension of the couplings $1\/G$ and $\\Lambda\/G$ that multiply the operators $\\sqrt{-g}$ and $\\sqrt{-g} R$. This occurs because the transformation between these couplings is non-singular, as explained in~\\cite{Percacci:2007sz}. On the other hand, at the Gaussian fixed point the transformation between the couplings is singular, and the dimensions change accordingly.}, while in the IR they become the canonical -2 and 2 respectively~\\cite{Litim:2012vz}. The fact that all couplings are attracted to the fixed point in the UV~\\cite{Codello:2008vh} is instead an artifact of the one-loop approximation. Indeed, more sophisticated FRG computations yield a fixed point with a three-dimensional critical surface~\\cite{Benedetti:2009rx}.\n\nThe flow can be solved analytically in terms of the deformations $\\delta \\lambda \\, , \\, \\delta \\omega$ from the UV fixed point, and yields the closed-form solution\n\\begin{eqaed}\\label{eq:marginal_flow}\n    \\lambda(t) & = \\frac{\\delta \\lambda}{1 - \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t} \\,,\n    \\\\\n    \\omega(t) & = \\frac{\\omega_- - \\, \\omega_+ \\, \\left(1 + \\frac{\\Delta}{\\delta \\omega}\\right) \\left(1 - \\, \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t \\right)^{\\frac{7\\sqrt{6049}}{399}}}{1 - \\, \\left(1 + \\frac{\\Delta}{\\delta \\omega}\\right) \\left(1 - \\, \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t \\right)^{\\frac{7\\sqrt{6049}}{399}}} \\,,\n    \\\\\n    \\theta(t) & = \\frac{56}{171} + \\frac{\\delta \\theta}{1 - \\frac{133}{160\\pi^2} \\, \\delta \\lambda \\, t} \\,,\n\\end{eqaed}\nwhere $\\Delta \\equiv \\omega_+ - \\omega_-$. The vector field generating this flow is displayed in fig.~\\ref{fig:flowlambdaomega} in the $(\\omega,\\lambda)$ subspace and in fig.~\\ref{fig:flowMC} with various 3D plots. Let us observe that the UV completeness of the trajectory requires $\\delta \\lambda > 0$, and that the IR flow ends at $t = t_\\text{IR} \\equiv \\frac{160\\pi^2}{133 \\delta \\lambda}$. However, since $\\delta \\lambda \\ll 1$ this RG time is parametrically large, and one can reliably extract the perturbative IR behavior for the Wilson coefficients. Furthermore, reaching a physical IR regime with $\\widetilde{G} \\to 0^+$ requires that $\\delta \\widetilde{G} \\, , \\, \\delta \\omega < 0$, in order that their flows remain between the UV and IR fixed-point values avoiding runaway. The flow of the relevant deformations from the fixed point is shown in Fig.~\\ref{fig:flow-reldeformations}.\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.7]{\"Images\/FlowLO\".pdf}\n\\caption{Flow of one-loop quadratic gravity in the $(\\omega,\\lambda)$ subspace. The arrows point toward the IR. In this setting, two non-trivial fixed points are present, and the Reuter fixed point is the one with the smaller absolute value~\\cite{Codello:2008vh}. \\label{fig:flowlambdaomega}}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.4]{\"Images\/Flow3D\".pdf}\\\\\\belowbaseline[0pt]{\n\\includegraphics[scale=0.45]{\"Images\/3Dplot-marg-4\".pdf}}~$\\qquad$\\belowbaseline[0pt]{\\includegraphics[scale=0.47]{\"Images\/3Dplot-marg-2\".pdf}}\n\\caption{Flow of the classically marginal couplings $(\\omega,\\lambda,\\theta)$ in one-loop quadratic gravity. The arrows point towards the IR, and different viewpoints are shown to better visualize the flow. The color coding of the arrows is identical to that of fig.~\\ref{fig:flowlambdaomega}.\\label{fig:flowMC}}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\includegraphics[scale=0.68]{\"Images\/FlowRDomlam\".pdf}\\includegraphics[scale=0.68]{\"Images\/FlowRDglambda\".pdf}\n\\caption{Flow of the relevant deformations from the fixed point. The left panel depicts the flow in the $(\\omega,\\lambda)$ subspace. The right panel depicts the flow in the $(G, \\Lambda)$ subspace where the classically marginal couplings have been set to the UV fixed point. The arrows indicate the the flow from the UV to the IR.\\label{fig:flow-reldeformations}}\n\\end{figure}\n\nSubstituting the expressions of eq.~\\eqref{eq:marginal_flow} in eq.~\\eqref{eq:betas}, one can then solve the remaining flow equations numerically varying the initial conditions, or, equivalently, the deformations $\\delta \\omega, \\delta \\lambda, \\delta \\widetilde{G} , \\delta \\widetilde{\\Lambda}$ from the UV fixed point. The RG flow then drives the running couplings to the weakly coupled IR, where the running couplings $g_C$ and $g_R$, defined in eq.~\\eqref{eq:wilson_coeffs}, behave logarithmically (linearly in $t$) as $t \\to t_\\text{IR}^-$. This result is consistent with perturbative computations, and the resulting asymptotic expressions read\n\\begin{eqaed}\\label{eq:log_running}\n    g_C(t) & \\sim \\frac{1}{2\\delta \\lambda} - \\frac{133}{320\\pi^2} \\, t \\, , \\\\\n    g_R(t) & \\sim - \\, \\frac{\\omega_-}{3\\delta \\lambda} + \\frac{133}{480\\pi^2} \\, \\omega_- \\, t \\, .\n\\end{eqaed}\nIn order to extract the physical IR parameters, we shall identify the (square of the) RG scale $k^2$ with the covariant Laplacian\/d'Alembertian $\\Box$. In order to eliminate the arbitrary reference scale $k_0$ that defines the initial condition for the RG flow, one can express every quantity in units of the IR Planck mass\\footnote{Notice that our convention for the Planck mass differs from the more widespread ``reduced'' Planck mass $\\widehat{M}_\\text{Pl}^{-2} = 8\\pi G$.} $M_\\text{Pl}^{-2} = G$. To this end, since $e^{2t} \\, \\widetilde{G}(t) \\to \\widetilde{G}_0$ tends to a constant in the IR, one can evaluate the running Wilson coefficients of eq.~\\eqref{eq:log_running} replacing $t \\to - \\, \\frac{1}{2} \\log \\widetilde{G}(t)$, so that\n\\begin{eqaed}\\label{eq:t_planck_sub}\n    \\log \\frac{M_\\text{Pl}}{k} & = \\log \\frac{M_\\text{Pl}}{k_0} + t \\\\\n    & \\sim - \\, \\frac{1}{2} \\log \\widetilde{G}(t) \\\\\n    &  \\sim - \\, \\frac{1}{2} \\log \\widetilde{G}_0 + t \\, .\n\\end{eqaed}\nSince $\\widetilde{G}_0$ can be extracted from the numerical solution of eqs.~\\eqref{eq:betas}, identifying \n\\begin{eqaed}\\label{eq:form_factor_sub}\n    \\log \\frac{k^2}{M_\\text{Pl}^2} \\longrightarrow \\log \\frac{\\Box}{M_\\text{Pl}^2} \\, ,\n\\end{eqaed}\naccording to the preceding considerations, one can reconstruct an effective action of the form\n\\begin{eqaed}\\label{eq:form_factor_eft}\n    \\Gamma = \\int d^4x \\sqrt{g} \\, \\left(\\frac{2\\Lambda - R}{16\\pi G} + g_C \\, C^2 + g_R \\, R^2 + b_C \\, C \\log \\frac{\\Box}{M_\\text{Pl}^2} \\, C + b_R \\, R \\log \\frac{\\Box}{M_\\text{Pl}^2} \\, R \\right) \\, , \n\\end{eqaed}\nwhere Weyl-tensor contraction is understood. The appearance of non-local form factors resonates with the considerations in~\\cite{Knorr:2019atm, Draper:2020bop, Draper:2020knh}. While we shall neglect them in the following, the presence of form factors of this type seems largely consistent with preceding results~\\cite{Riegert:1984kt, Deser:1996na, Erdmenger:1996yc, Erdmenger:1997gy, Deser:1999zv, Bautista:2017enk} (see also~\\cite{Donoghue:2015nba} for a discussion of logarithmic form factors). Note that, despite their behavior at low energies, one expects a resummation of such non-local form factors to yield a result that is both well-defined and subleading in the IR compared to the local terms~\\cite{Draper:2020bop}\\footnote{One exception could be a non-local form factor of the type $\\sim 1\/\\Box$, as discussed in~\\cite{Belgacem:2017cqo,Knorr:2018kog}.}. Furthermore, they do not contribute to the scalar potential that we shall discuss in Section~\\ref{sec:dsc_constraints} in the context of the dSC. Notwithstanding the importance of form factors in establishing a non-local behavior of gravity, we would like to understand which values of the three dimensionless combinations\n\\begin{eqaed}\\label{eq:IR_params}\n    G\\Lambda \\, , \\quad g_C \\, , \\quad g_R\n\\end{eqaed}\nare allowed starting from any initial condition, \\emph{i.e.}, any perturbation of the asymptotically safe UV fixed point along UV-attractive directions. To this end, we have evaluated numerically these combinations in the IR, implementing the substitution of eq.~\\eqref{eq:t_planck_sub}. The following plots highlighting the swampland constraints, the IR limits of asymptotically safe RG trajectories, as well as the final intersection between the allowed regions, will pertain to the $(G\\Lambda,g_C,g_R)$ theory space.\n\nTo conclude this section, let us collect a few words of caution regarding the one-loop approximation. In general, in the context of gravity, one expects it to be only reliable in the IR, despite the appearance of a UV fixed point outside of the perturbative regime. The methods of the functional RG have been employed, both in earlier~\\cite{Benedetti:2009rx,Benedetti:2009gn} and recent~\\cite{Knorr:2021slg} efforts, to obtain non-perturbative flow equations in the quadratic truncation, but applying our method to extract the allowed region of parameter space in the IR entails highly involved and unstable numerical analysis. In order to circumvent these obstacles, and address the problem in a more quantitative fashion, a natural first step would entail performing novel FRG computations. The simplest relevant setting would include the most general quadratic truncation coupling the electromagnetic field to gravity, which, while daunting, appears feasible via the methods that have been very recently introduced in~\\cite{Knorr:2021slg} to study the purely gravitational sector. In light of these (and other related) issues, in this work we have focused on the one-loop approximation as a proof of principle, with the hope of uncovering some instructive general lessons from the results that we are now about to present.\n\nFinally, let us stress that, although quadratic truncations of the gravitational action are typically associated with a loss of physical unitarity~\\cite{Stelle:1977ry}, the Stelle ghost could be a truncation artifact~\\cite{Platania:2020knd}. Integrating out quantum fluctuations could lead to well-behaved, unitary scattering amplitudes~\\cite{Draper:2020bop,Draper:2020knh}, as explicit computations seem to suggest~\\cite{Bonanno:2021squ}. This issue was also discussed within the setting explored in this paper in~\\cite{Niedermaier:2009zz, Niedermaier:2010zz}.\n\n\\section{Results}\\label{sec:results}\n\nLet us now describe in detail our results on the allowed values of physical IR parameters that we have obtained from the calculations outlined in the preceding section, along with the swampland constraints that we have discussed in sect.~\\ref{sec:swampland}.\n\n\\subsection{Infrared limit of asymptotically safe RG trajectories in one-loop quadratic gravity}\\label{sec:IR_space}\n\nIn order to uncover the space of physical parameters appearing in the (local sector of the) effective action of eq.~\\eqref{eq:form_factor_eft}, we have sampled the space of allowed deformations $(\\delta \\omega, \\delta \\lambda, \\delta \\widetilde{G} , \\delta \\widetilde{\\Lambda})$ from the UV fixed point, and extracted the resulting IR values of the parameters in eq.~\\eqref{eq:IR_params} evaluating the flow of $\\widetilde{G}$ and $\\widetilde{\\Lambda}$ for a suitably large RG time $t \\approx 30$, exploiting the rapid convergence of the combination in eq.~\\eqref{eq:t_planck_sub}. The resulting values for $\\widetilde{G}\\widetilde{\\Lambda} = G\\Lambda$, or equivalently $\\Lambda\/M_\\text{Pl}^2$, span a wide range of values, of the order of $10^5$ for the region of initial deformations that we have explored. Moreover, the closed-form flow that one obtains setting the classically marginal couplings to their fixed-point values spans the whole real axis~\\cite{Codello:2008vh}. We are thus led to conclude that the allowed (IR) values of $G\\Lambda$ are unrestricted. On the other hand, the values of $g_C$ and $g_R$ appear to lie on the line\n\\begin{eqaed}\\label{eq:marginal_line}\n    g_R \\approx - \\, 0.74655 + 3.64447 \\, g_C \\, ,\n\\end{eqaed}\nas depicted in fig.~\\ref{fig:IR2d-fit} and fig.~\\ref{fig:IR3d}. This result appears to be very robust upon increase of the sample size, and in particular for $10^6$ points the covariance matrix of the fit is of the order $\\mathcal{O}(10^{-8})$. Let us observe that, neglecting the intercept term, eq.~\\eqref{eq:marginal_line} follows from eq.~\\eqref{eq:log_running} as $t \\to t_\\text{IR}^-$, whereby $g_R \\sim - \\, 2\\omega_-\/3 \\, g_C$. Since we instead evaluate the IR couplings at a fixed, albeit sufficiently large, RG time, it is tempting to speculate that the intercept term in eq.~\\eqref{eq:marginal_line} is a correction arising from RG trajectories that approach the IR more slowly. Therefore, at least within the scope of our approximations, the presence of a UV fixed point appears to constrain the allowed physical coefficients in eq.~\\eqref{eq:IR_params} to a specific plane, and we shall now compare this result to the constraints arising from the swampland conjectures that we have discussed in sect.~\\ref{sec:swampland}. \n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.5]{\"Images\/LineEqs\".pdf}\n\\caption{The line of equation $g_R = - \\, 0.74655 + 3.64447 \\, g_C$ fitting the IR values obtained from the flow of eq.~\\eqref{eq:betas} sampling UV initial conditions. The covariance matrix evaluates to $\\mathcal{O}(10^{-8})$ with~$10^6$ data points.}\\label{fig:IR2d-fit}\n\\end{figure}\n\n\\begin{figure}[t!]\n$\\hspace{-0.2cm}$\\includegraphics[scale=0.53]{\"Images\/Plot3D-1\".pdf}$\\hspace{-0.5cm}$\\includegraphics[scale=0.53]{\"Images\/Plot3D-2\".pdf}\\\\\n\\centering\\includegraphics[scale=0.55]{\"Images\/Plot3D-3\".pdf}\n\\caption{Plots depicting the IR endpoints of asymptotically safe RG trajectories. The points lie on the plane of equation $g_R = - \\, 0.74655 + 3.64447 \\, g_C$. The values of $G\\Lambda$ span a vast range, which, together with the closed-form flow of~\\cite{Codello:2008vh} depicted in fig.~\\ref{fig:flow-reldeformations}, leads us to infer that they are unrestricted.}\\label{fig:IR3d}\n\\end{figure}\n\n\\subsection{Constraints on quadratic gravity from WGC}\\label{sec:wgc_constraints}\n\nAs we have discussed in sect.~\\ref{sec:wgc_intro}, the WGC entails positivity bounds for the Wilson coefficients of the higher-derivative corrections to Einstein gravity. Since these bounds involve charged particles and black holes, higher-derivative couplings of a $U(1)$ gauge field ought to be included, although the resulting RG flow is extremely involved technically and has not been computed hitherto. On the other hand, the considerations of~\\cite{Charles:2017dbr, Charles:2019qqt, Cano:2021tfs}, based on electromagnetic duality, show that one can still make use of our results to constrain higher-derivative corrections in a \\emph{duality-invariant} scenario using the WGC. To this end, expressing the higher-curvature in terms of the $c_i$ coefficients of eq.~\\eqref{eq:eft_corr}, the (family of) positivity bound(s) of~\\cite{Cheung:2018cwt} reads\n\\begin{eqaed}\\label{eq:positivity_bound}\n    (1-\\xi)^2\\,c_0+20\\xi c_3-5\\xi(1-\\xi)(2c_3)>0\n\\end{eqaed}\nwhere $\\xi\\equiv\\sqrt{1-Q^2\/M^2}$ is the extremality parameter of Reissner-Nordstr\\\"{o}m black holes with mass $M$ and charge $Q$, $0<\\xi<1\/2$ for black holes with positive specific heat and\n\\begin{eqaed}\\label{eq:c0_coeff}\n    c_0 \\equiv c_2 + 4 \\, c_3 \\, .\n\\end{eqaed}\nIn terms of the couplings in eq.~\\eqref{eq:quadratic_lagrangian}, the bound of eq.~\\eqref{eq:positivity_bound} takes the simpler form\n\\begin{eqaed}\\label{eq:simpler_wgc_bound}\n    \\frac{1}{\\lambda}(10\\, \\theta \\, (\\xi +1) \\xi +6 \\xi ^2+3 \\xi +1)>0 \\, ,\n\\end{eqaed}\nwhich holds for~$\\lambda>0$ provided that~$\\xi>0$ (which is always satisfied by the extremality parameter) and that~$\\theta>0$. As we have discussed in sect.~\\ref{sec:one-loop}, the latter condition is fulfilled  if $\\delta \\theta > 0$, since $\\delta \\lambda > 0$ in eq.~\\eqref{eq:marginal_flow}. Hence, the (duality-invariant) WGC constrains~$\\lambda > 0$, which is included by the analysis of the preceding section and does not entail additional conditions. Let us observe that, although $\\theta$ encodes the coupling of the Gauss-Bonnet invariant, it contributes to the entropy of a black hole~\\cite{Myers:1988ze,Myers:1998gt,Clunan:2004tb} even in four dimensions, where it is topological. It does not contribute in the limit $\\xi \\ll 1$, since the resulting bound also describes the positivity of the extremality ratio~\\cite{Kats:2006xp,Charles:2019qqt}.\n\n\\subsection{Constraints on quadratic gravity from dS and TC conjectures}\\label{sec:dsc_constraints}\n\nLet us now discuss the constraints arising from the dSC and the TCC. As we have anticipated in sect.~\\ref{sec:dsc_tcc_intro}, we shall focus on the bounds that the dSC and the TCC entail for the scalar potential that arises from the higher-derivative corrections of eq.~\\eqref{eq:form_factor_eft}. In order to extract the potential proper, we shall concern ourselves with the local sector of the theory, neglecting the form factors and the Weyl term, which vanishes on cosmological backgrounds.\n\nFollowing the standard procedure to obtain inflaton potentials from $F(R)$ Lagrangians (see, \\emph{e.g.},~\\cite{inflationaris,Platania:2019qvo}), one begins from $R^2$ gravity with a cosmological constant,\n\\begin{eqaed}\\label{eq:our_f}\n    F(R)=\\frac{1}{16\\pi G}\\left(R-2\\Lambda + \\frac{R^2}{6m^2}\\right) \\, ,\n\\end{eqaed}\nwhere the coupling $g_R$ is related to the inflaton mass according to\n\\begin{eqaed}\\label{eq:grm2}\n    g_R = - \\, \\frac{M_\\text{Pl}^2}{(8\\pi)\\cdot 12m^2} \\, .\n\\end{eqaed}\nOne then arrives at the inflaton potential\n\\begin{eqaed}\\label{eq:scalar_pot}\n    V(\\phi)=\\frac{M_\\text{Pl}^2}{8\\pi}\\,e^{-2\\sqrt{\\frac{2}{3}}\\frac{\\phi}{M_\\text{Pl}}} \\left(\\frac{3 m^2}{4}\\left(e^{\\sqrt{\\frac{2}{3}}\\frac{\\phi}{M_\\text{Pl}}}-1\\right)^2+\\Lambda \\right) \\, .\n\\end{eqaed}\nIn order to retain compatibility with the EFT, we shall consider field values in eq.~\\eqref{eq:field_range} around $\\phi \\ll M_\\text{Pl}$, since it corresponds to small curvatures. Indeed, the procedure to obtain inflationary potentials from quadratic gravity yields $\\phi = \\sqrt{\\frac{3}{2}} \\, M_\\text{Pl} \\log\\left( 1 + \\mathcal{O}(M_\\text{Pl}^{-2})\\right)$~\\cite{inflationaris,Platania:2019qvo}. One can then study dSC and TCC constraints of eqs.~\\eqref{eq:dsc_basic},~\\eqref{eq:ref_dsc} and~\\eqref{eq:tcc_basic} numerically varying the $\\mathcal{O}(1)$ constants $c$ and $f$, imposing that the bounds be satisfied for all $\\phi$ in the range allowed by eq.~\\eqref{eq:field_range}. The resulting regions are highlighted in fig.~\\ref{fig:TCC1} and in fig.~\\ref{fig:TCC2}, where each panel corresponds to a particular value of $f$ and consists of two plots which display the bounds in the $(m^2,\\Lambda)$ (left panels) and $(g_R,G\\Lambda)$ (right panels) planes. Due to the inverse relation in eq.~\\eqref{eq:grm2} between $m^2\/M_{\\text{Pl}}^2$ and $g_R$, the linear bounds in the~$(m^2, \\Lambda)$ plane translate into hyperbolas in the~$(g_R,G\\Lambda)$ plane. Note that whether the dimensionless minimum $\\phi_{\\mathrm{min}}\/M_{\\mathrm{Pl}}$, which exists for $\\Lambda \\,\/{m^2} >-3\/4$, falls inside the interval $(-f,+f)$ depends on the ratio $\\Lambda\/m^2$. In particular, the minimum falls outside the interval for $f<|\\phi_\\mathrm{min}\/M_\\mathrm{Pl}|$. Consequently, even if $V(\\phi_\\text{min})$ were to be positive for some $\\Lambda$ and $m^2$, this would not necessarily violate the dSC\/TCC bounds for fixed values of $f$ and $c$. Consequently, the bounds displayed in fig.~\\ref{fig:TCC1} and in fig.~\\ref{fig:TCC2} are non-trivially affected by eq.~\\eqref{eq:field_range}, and by the specific values of $f$ and $c$. For instance, it is worth noticing that smaller values of $f$ entail smaller field intervals where the dSC\/TCC bounds are to be satisfied. Thus, the bounds are less stringent, and the allowed region bigger. Similarly, the bound is more restrictive for higher values of $c$. In particular, fig.~\\ref{fig:TCC2} depicts the bounds derived from $c = \\sqrt{2\/3}$, the value pertaining to the TCC. While our analysis cannot probe the TCC in the large-excursion regime, where it differs substantially from the dSC, the additional considerations of~\\cite{Andriot:2020lea, Rudelius:2021oaz} point to a deeper role of this value of $c$ which could manifest itself, in the low-curvature regime at stake, in further investigations of swampland bounds and\/or dimensional reduction.\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCf01-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCf01-gr\".pdf}\\\\\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCf05-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCf05-gr\".pdf}\\\\\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCf1-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCf1-gr\".pdf}\n\\caption{dSC\/TCC constraints for $f=0.1$ (top panel), $f=0.5$ (central panel) and $f=1$ (bottom panel), and various values of $c$. Due to the inverse relation in eq.~\\eqref{eq:grm2} between the inflaton mass in Planck units $m^2\/M_{\\text{Pl}}^2$ and the coupling $g_R$, the linear bounds in the~$(m^2, \\Lambda)$ plane translate into hyperbolas in the~$(g_R,G\\Lambda)$ plane. \\label{fig:TCC1}}\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.65]{\"Images\/TCCc23-m2\".pdf}$\\,\\,$\\includegraphics[scale=0.65]{\"Images\/TCCc23-gr\".pdf}\n\\caption{TCC constraints, corresponding to $c=\\sqrt{2\/3}$, for various values of $f$. The bounds are not qualitatively different from to the dSC bounds displayed in fig.~\\ref{fig:TCC1}.\\label{fig:TCC2}}\n\\end{figure}\n\n\\subsection{Intersections of allowed regions: compatibility of asymptotic safety with dS, TC and WG conjectures}\\label{sec:intersections}\n\nWe are now ready to collect the results that we have discussed in the preceding sections, and to visualize the intersection between the different allowed regions. Within (an extrapolation of the) one-loop approximation, asymptotic safety of the RG flow constrains the physical IR parameters of eq.~\\eqref{eq:IR_params} to lie on the plane of eq.~\\eqref{eq:marginal_line}. On the other hand, while the (duality-invariant) WGC does not entail any additional constraint, the dSC\/TCC conditions for the inflaton potential place constraints on the cosmological constant and the inflaton mass. One can plot the intersections for any values of the $\\mathcal{O}(1)$ constants $c$ and $f$, and fig.~\\ref{fig:intersections3D}, the main result of our work, refers to the representative choice $c=f=1$. Albeit difficult to visualize, there is in general a region of the asymptotically safe plane that appears compatible with the swampland constraints that we have investigated. One can straightforwardly verify that the same conclusion is reached for different values of $f$ and $c$. Our findings, based on the quadratic one-loop approximation, thus point at a non-trivial compatibility between the conditions for UV-completeness dictated by asymptotic safety and some of the most relevant swampland conjectures. Consequently, it also points at the possibility, partially supported by~\\cite{Basile:2020dzh,Basile:2021krk}, of a connection between the frameworks of asymptotic safety and string theory~\\cite{deAlwis:2019aud}. Within this picture, field-theoretical asymptotic safety would serve as a ``pivot'' for the RG flow from string theory to low-energy gravity, in the sense that below a certain scale the flow of string theory toward the IR closely approaches a field-theoretical trajectory controlled by a UV fixed point.\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[scale=0.5]{\"Images\/Intersections-3D\".pdf}\\includegraphics[scale=0.5]{\"Images\/Intersections-3D-2\".pdf}\\\\\\includegraphics[scale=0.5]{\"Images\/Intersections-3D-3\".pdf}\n\\caption{Intersections between the regions allowed by asymptotic safety, the WGC and the dSC\/TCC for the representative values $c=f=1$. The WGC bound corresponds to the yellow region, while the dSC\/TCC bound corresponds to the blue region. The green region depicts the space of IR parameters spanned by asymptotically safe trajectories, which lies within the region allowed by the WGC.}\\label{fig:intersections3D}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this paper we have analyzed the intersection of consistency conditions for Wilson coefficients of gravitational EFTs, combining the constraints of asymptotic safety and swampland conjectures. In particular, in sect.~\\ref{sec:one-loop} we have employed a systematic method to extract the  hypersurface of allowed IR parameters stemming from UV-complete RG trajectories in gravitational theories by randomly sampling its relevant deformations, and we have applied this technique to the flow equations stemming from one-loop quadratic gravity~\\cite{Codello:2008vh}. Despite expecting that this approximation be reliable in the IR, the resulting RG flow exhibits a UV non-Gaussian fixed point, consistently with more refined functional RG computations~\\cite{Percacci:2017fkn, Reuter:2019byg, Pawlowski:2020qer}. However, the dimension of its critical surface is larger than what is suggested by the functional RG~\\cite{Benedetti:2009rx,Benedetti:2009gn,Falls:2020qhj,Knorr:2021slg}. As we have discussed in detail in sect.~\\ref{sec:results}, our findings suggest that the requirement that the RG flow be asymptotically safe constrains the physical parameters to lie on a plane, which we have determined, within our one-loop framework, to a precision of order $\\mathcal{O}(10^{-8})$. The values of the cosmological constant in Planck units seems not to be restricted by these considerations, while the classically marginal couplings lie on a line. \n\nIn sect.~\\ref{sec:wgc_constraints} and sect.~\\ref{sec:dsc_constraints} we have investigated the constraints on the Wilson coefficients arising from the weak gravity conjecture (WGC), the de Sitter conjecture (dSC) and the trans-Planckian censorship conjecture (TCC). In particular, the WGC does not entail additional bounds and is compatible with the UV-complete RG trajectories, while the dSC\/TCC bounds are more restrictive. To wit, the Starobinsky-like scalar potential stemming from the (local sector of the) effective action involves the ratio of the cosmological constant to the (squared) inflaton mass, and therefore the corresponding bounds place constraints on the dimensionless ratios $\\Lambda\/M_\\text{Pl}^2$ and $m\/M_\\text{Pl}$. These bounds depend on some dimensionless $\\mathcal{O}(1)$ constants, which we have varied to some extent in our analysis, and generally trace out a region in the plane allowed by asymptotic safety. While we expect that the qualitative results be unaffected by improving the truncation scheme, at least to some extent, it would be interesting to investigate the quantitative deviations in this respect.\n\nWhile in this work we have focused on the local sector of the effective action, our computation also yields the coefficients of non-local logarithmic form factors, akin to those arising from non-local heat kernel computations~\\cite{Barvinsky:1987uw, Barvinsky:1990up, Barvinsky:1990uq, Barvinsky:1993en, Avramidi:1990ap, Codello:2012kq}. It would be interesting to explore their consequences and their role within asymptotically safe gravity~\\cite{Knorr:2019atm, Draper:2020bop, Draper:2020knh} and their connection to massive matter fields~\\cite{Ohta:2020bsc}.\n\nAll in all, our results suggest that swampland constraints can be compatible with restrictions coming from UV completeness of the RG flow, but in a non-trivial fashion: the allowed parameter space is restricted to a non-trivial intersection. In retrospect, one could have expected this result on the grounds that some swampland criteria purport to be necessary conditions for UV completeness that cannot be derived from  purely field-theoretical considerations, and thus they could constrain further the parameter space compatible with asymptotic safety. On the other hand, the one-loop approximation that we have studied already features the appearance of non-local form factors. In general, non-locality at the level of the effective action is a feature of any standard (local) QFT, and thus it is in principle unrelated to possible fundamental non-localities in the bare (fixed-point) action. Precisely how the notion of (non-)locality is realized in quantum gravity is an open and intriguing question, partly related to the problem of observables~\\cite{Donnelly:2015hta, Rejzner:2016yuy, Donnelly:2016rvo, Klitgaard:2017ebu, Rudelius:2021azq}. However, a number of semi-classical considerations~\\cite{Giddings:1992hh, Susskind:1993if, Almheiri:2012rt, Giddings:2012gc, Dvali:2014ila, Keltner:2015xda, Mann:2015luq} point to the breaking of the familiar concept of locality microscopically. Whether asymptotically safe gravity is realized by a bare action polynomial in derivatives (and thus ``local'' in some sense) is not established yet. Should fundamental non-locality turn out to emerge as a feature of asymptotic safety, this would strengthen its potential connections with the frameworks of non-local gravity~\\cite{Modesto:2011kw, Modesto:2017sdr, Buoninfante:2018mre, Buoninfante:2018xiw} and string theory~\\cite{Giddings:2006vu}.\n\nDue to the nature of our approximations, this work constitutes only a first step toward determining whether the asymptotic safety scenario is compatible with the peculiar behavior and UV\/IR mixing that gravity could exhibit already at the semi-classical level due to black holes (see~\\cite{Buoninfante:2021ijy} for a very recent discussion on their validity and limitations), or with general indications from string theory. In particular, a possible connection between asymptotically safe gravity and string theory has been conjectured in~\\cite{deAlwis:2019aud}, and it is tempting to speculate that it could explain our findings. Computations combining the functional renormalization group techniques~\\cite{Dupuis:2020fhh} with symmetries of string theory~\\cite{Veneziano:1991ek, Meissner:1991zj, Meissner:1996sa, Hohm:2015doa, Hohm:2019ccp, Hohm:2019jgu} have provided preliminary evidence in favour of this scenario~\\cite{Basile:2020dzh,Basile:2021krk}. This possibility extends to the more general notion of ``effective asymptotic safety''~\\cite{Held:2020kze}, and swampland bounds could further constrain which RG flows controlled by the ``effective'' fixed point are closely approached by the RG flow arising from the proper UV completion in the IR.\n\nMost prominently, the absence of continuous global symmetries\\footnote{The fate of global discrete symmetries has been investigated in the context of asymptotically safe gravity in~\\cite{Eichhorn:2020sbo, Ali:2020znq}.} is supported by a variety of arguments from black-hole physics, string theory and holography~\\cite{Misner:1957mt, Banks:1988yz, Kallosh:1995hi, Polchinski:2003bq, Banks:2010zn, Harlow:2018tng, McNamara:2019rup}, and it would be interesting to explore this foundational issue further in the direction that we have outlined in this paper.\n\n\\section*{Acknowledgements}\n\nThe authors would like to thank F. Saueressig for insighful discussions and B. Knorr for feedback on the manuscript. The authors thank also B. Holdom for spotting a typo.\n\nThe work of I.B. is supported by the Fonds de la Recherche Scientifique - FNRS under Grants No. F.4503.20 (\"HighSpinSymm\") and T.0022.19 (\"Fundamental issues in extended gravitational theories\"). A.P. acknowledges support by Perimeter Institute for Theoretical Physics.  Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities.\n\n\\bibliographystyle{JHEP}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\\protect\\setcounter{equation}{0}\n\n\n\\subsection{}\nThe quantum systems studied in this paper are obtained by coupling a\ncertain number (finite or infinite) of $N$-body systems. A\n(standard) $N$-body system consists of a fixed number $N$ of\nparticles which interact through $k$-body forces which preserve $N$\n(arbitrary $1\\leq k \\leq N$). The many-body type interactions\ninclude forces which allow the system to make transitions between\nstates with different numbers of particles. These transitions are\nrealized by creation-annihilation processes as in quantum field\ntheory.\n\nThe Hamiltonians we want to analyze are rather complex objects and\nstandard Hilbert space techniques seem to us inefficient in this\nsituation. Our approach is based on the observation that the\n$C^*$-algebra $\\mathscr{C}$ generated by a class of physically interesting\nHamiltonians often has a quite simple structure which allows one to\ndescribe its quotient with respect to the ideal of compact operators\nin rather explicit terms \\cite{GI1,GI2}.  From this one can deduce\ncertain important spectral properties of the Hamiltonians.  We refer\nto $\\mathscr{C}$ as the \\emph{Hamiltonian algebra} (or $C^*$-algebra of\nHamiltonians) of the system. \n\nThe main difficulty in this algebraic approach is to isolate the\ncorrect $C^*$-algebra. This is especially problematic in the present\nsituations since it is not a priori clear how to define the\ncouplings between the various $N$-body systems but in very special\nsituations.  It is rather remarkable that the $C^*$-algebra\ngenerated by a small class of elementary and natural Hamiltonians\nwill finally prove to be a fruitful choice.  These elementary\nHamiltonians are analogs of the Pauli-Fierz Hamiltonians.\n\n\nThe purpose of the preliminary Section \\ref{s:euclid} is to present\nthis approach in the simplest but physically important case when the\nconfiguration spaces of the $N$-body systems are Euclidean\nspaces. We start with a fundamental example, the standard $N$-body\ncase. Then we describe the many-body formalism in the Euclidean case\nand we state our main results on the spectral analysis of the\ncorresponding Hamiltonians.\n\nThere is one substantial simplification in the Euclidean case: each\nsubspace has a canonical supplement, the subspace orthogonal to\nit. This plays a role in the way we present the framework in Section\n\\ref{s:euclid}.  However, the main constructions and results do not\ndepend on the existence of a supplement but to see this requires\nmore sophisticated tools from the theory of crossed product\n$C^*$-algebras and Hilbert $C^*$-modules which are not apparent in\nthis introductory part.  In the rest of the paper we consider\nmany-body type couplings of systems whose configuration space is an\narbitrary abelian locally compact group.  One of the simplest\nnontrivial physically interesting cases covered by this framework is\nthat when the configuration spaces of the $N$-body systems are\ndiscrete groups, e.g.  discretizations $\\mathbb{Z}^D$ of $\\mathbb{R}^D$.\n\n\\subsection{}\nWe summarize now the content of the paper.  Section \\ref{s:euclid}\nstarts with a short presentation of the standard \\mbox{$N$-body}\nformalism, the rest of the section being devoted to a rather\ndetailed description of our framework and main results in the case\nwhen the configuration spaces of the $N$-body subsystems are\nEuclidean spaces. These results are proven in a more general and\nnatural setting in the rest of the paper.  In Section \\ref{s:grad}\nwe recall some facts concerning $C^*$-algebras graded by a\nsemilattice $\\mathcal{S}$ (we take here into account the results of Athina\nMageira's thesis \\cite{Ma}) and then we present some results on\n$\\mathcal{S}$-graded Hilbert $C^*$-modules. This notion, due to Georges\nSkandalis \\cite{Sk}, proved to be very natural and useful in our\ncontext: thanks to it many results can be expressed in a simple and\nsystematic way thus giving a new and interesting perspective to the\nsubject (this is discussed in more detail in \\cite{DG4}). The heart\nof the paper is Section \\ref{s:grass}, where we define the many-body\nHamiltonian algebra $\\mathscr{C}$ in a general setting and prove that it is\nnaturally graded by a certain semilattice $\\mathcal{S}$. In Section\n\\ref{s:id} we give alternative descriptions of the components of\n$\\mathscr{C}$ which are important for the affiliation criteria presented in\nSection \\ref{s:af}, where we point out a large class of self-adjoint\noperators affiliated to the many-body algebra.  The $\\mathcal{S}$-graded\nstructure of $\\mathscr{C}$ gives then an HVZ type description of the\nessential spectrum for all these operators. The main result of\nSection \\ref{s:mou} is the proof of the Mourre estimate for\nnonrelativistic many-body Hamiltonians. Finally, an Appendix is\ndevoted to the question of generation of some classes of\n$C^*$-algebras by \"elementary\" Hamiltonians.\n\n\n\\subsection{Notations} \n\\label{ss:inotations}\nWe recall some notations and terminology.  If $\\mathcal{E},\\mathcal{F}$ are normed\nspaces then $L(\\mathcal{E},\\mathcal{F})$ is the space of bounded operators\n$\\mathcal{E}\\rightarrow\\mathcal{F}$ and $K(\\mathcal{E},\\mathcal{F})$ the subspace consisting of compact\noperators. If $\\mathcal{G}$ is a third normed space and $(e,f)\\mapsto ef$ is\na bilinear map $\\mathcal{E}\\times\\mathcal{F}\\to\\mathcal{G}$ then $\\mathcal{E}\\mathcal{F}$ is the linear\nsubspace of $\\mathcal{G}$ generated by the elements $ef$ with\n$e\\in\\mathcal{E},f\\in\\mathcal{F}$ and $\\mathcal{E}\\cdot\\mathcal{F}$ is its closure. If $\\mathcal{E}=\\mathcal{F}$\nthen we set $\\mathcal{E}^2=\\mathcal{E}\\cdot\\mathcal{E}$.  Two unusual abbreviations are\nconvenient: by \\emph{lspan} and \\emph{clspan} we mean ``linear\nspan'' and ``closed linear span'' respectively.  If $\\mathcal{A}_i$ are\nsubspaces of a normed space then $\\sum^\\mathrm{c}_i\\mathcal{A}_i$ is the clspan of\n$\\cup_i\\mathcal{A}_i$.  If $X$ is a locally compact topological space then\n$\\cc_{\\mathrm{o}}(X)$ is the space of continuous complex functions which tend to\nzero at infinity and $\\cc_{\\mathrm{c}}(X)$ the subspace of functions with compact\nsupport.\n\nBy \\emph{ideal} in a $C^*$-algebra we mean a closed self-adjoint\nideal. A $*$-homomorphism between two \\mbox{$C^*$-algebras} will be\ncalled \\emph{morphism}.  We write $\\mathscr{A}\\simeq\\mathscr{B}$ if the\n$C^*$-algebras $\\mathscr{A},\\mathscr{B}$ are isomorphic and $\\mathscr{A}\\cong\\mathscr{B}$ if they\nare canonically isomorphic (the isomorphism should be clear from the\ncontext).\n\n\nA self-adjoint operator $H$ on a Hilbert space $\\mathcal{H}$ is\n\\emph{affiliated} to a $C^*$-algebra $\\mathscr{A}$ of operators on $\\mathcal{H}$ if\n$(H+i)^{-1}\\in\\mathscr{A}$; then $\\varphi(H)\\in\\mathscr{A}$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$. If $\\mathscr{A}$ is the closed linear span of the\nelements $\\varphi(H)A$ with $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$ and $A\\in\\mathscr{A}$, we\nsay that $H$ is \\emph{strictly affiliated to $\\mathscr{A}$}.  The\n$C^*$-algebra generated by a set $\\mathscr{E}$ of self-adjoint operators is\nthe smallest $C^*$-algebra such that each $H\\in\\mathscr{E}$ is affiliated to\nit.\n\nWe now recall the definition of $\\mathcal{S}$-graded $C^*$-algebras\nfollowing \\cite{Ma2}. Here $\\mathcal{S}$ is a \\emph{semilattice}, i.e. a set\nequipped with an order relation $\\leq$ such that the lower bound\n$\\sigma\\wedge\\tau$ of each couple of elements $\\sigma,\\tau$ exists.\nWe say that a subset $\\mathcal{T}$ of $\\mathcal{S}$ is a \\emph{sub-semilattice} of\n$\\mathcal{S}$ if $\\sigma,\\tau\\in\\mathcal{T}\\Rightarrow\\sigma\\wedge\\tau\\in\\mathcal{T}$.  The\nset $\\mathscr{S}$ of all closed subgroups of a locally compact abelian group\nis a semilattice for the order relation given by set inclusion.  The\nsemilattices which are of main interest for us are (inductive limits\nof) sub-semilattices of $\\mathscr{S}$.\n\nA $C^*$-algebra $\\mathscr{A}$ is called \\emph{$\\mathcal{S}$-graded} if a linearly\nindependent family $\\{\\mathscr{A}(\\sigma)\\}_{\\sigma\\in\\mathcal{S}}$ of\n$C^*$-subalgebras of $\\mathscr{A}$ has been given such that\n$\\sum^\\mathrm{c}_{\\sigma\\in\\mathcal{S}}\\mathscr{A}(\\sigma)=\\mathscr{A}$ and\n$\\mathscr{A}(\\sigma)\\mathscr{A}(\\tau)\\subset\\mathscr{A}(\\sigma\\wedge\\tau)$ for all\n$\\sigma,\\tau$. The algebras $\\mathscr{A}(\\sigma)$ are the \\emph{components\n  of $\\mathscr{A}$}. It is useful to note that some of the algebras\n$\\mathscr{A}(\\sigma)$ could be zero. If $\\mathcal{T}$ is a sub-semilattice of\n$\\mathcal{S}$ and $\\mathscr{A}(\\sigma)=\\{0\\}$ for $\\sigma\\notin\\mathcal{T}$ we say that $\\mathscr{A}$\nis \\emph{supported} by $\\mathcal{T}$; then $\\mathscr{A}$ is in fact\n$\\mathcal{T}$-graded. Reciprocally, any $\\mathcal{T}$-graded $C^*$-algebra becomes\n$\\mathcal{S}$-graded if we set $\\mathscr{A}(\\sigma)=\\{0\\}$ for $\\sigma\\notin\\mathcal{T}$.\n\n\n\n\n\\subsection{Note}\nThe preprint \\cite{DG4} is a preliminary version of this paper. We\ndecided to change the title because the differences between the two\nversions are rather important: the preliminaries concerning the\ntheory of Hilbert $C^*$-modules and the role of the imprimitivity\nalgebra of a Hilbert $C^*$-module in the spectral analysis of\nmany-body systems are now reduced to a minimum; on the other hand,\nthe Euclidean case and the spectral theory of the corresponding\nHamiltonians are treated in more detail.\n\n\\vspace{3mm}\n\n\\begin{acknowledgement}{\\rm\nThe authors thank Georges Skandalis for very helpful suggestions and\nremarks.}\n\\end{acknowledgement}\n\n\n\\section{Euclidean framework: main results}\n\\label{s:euclid}\n\\protect\\setcounter{equation}{0}\n\n\\subsection{The Hamiltonian algebra of a standard $N$-body system}\n\\label{ss:inb}\n\nConsider a system of $N$ particles moving in the physical space\n$\\mathbb{R}^d$. In the nonrelativistic case the Hamiltonian is of the form\n\\begin{equation}\\label{eq:inonrel}\nH={\\textstyle\\sum_{j=1}^N P_j^2\/2m_j + \\sum_{j=1}^N V_j(x_j)+\n\\sum_{j<k} V_{jk}(x_j-x_k)}\n\\end{equation}\nwhere $m_1,\\dots,m_N$ are the masses of the particles,\n$x_1,\\dots,x_N\\in\\mathbb{R}^d$ their positions, and $P_j=-i\\nabla_{x_j}$\ntheir momenta. In the simplest situation the potentials $V_j,V_{jk}$\nare real continuous functions with compact support on $\\mathbb{R}^d$. The\nstate space of the system is the Hilbert space $L^2(X)$ with\n$X=(\\mathbb{R}^d)^N$.\n\nLet $P=(P_1,\\dots,P_N)$, this is a set of commuting self-adjoint\noperators on $L^2(X)$ and so $h(P)$ is a well defined self-adjoint\noperator for any real Borel function $h$ on $X$. In what follows we\nreplace the kinetic energy part $\\sum_j P_j^2\/2m_j$ in\n\\eqref{eq:inonrel} by an operator $h(P)$ with $h$ continuous and\ndivergent at infinity. Denote $x=(x_1,\\dots,x_N)$ the points of $X$\nand let us consider the linear subspaces of $X$ defined as follows:\n$X_j=\\{x\\in X\\mid x_j=0\\}$ if $1\\le j \\le N$ and $X_{jk}=\\{x\\in\nX\\mid x_j=x_k\\}$ for $j<k$.  Let $\\pi_j$ and $\\pi_{jk}$ be the\nnatural maps of $X$ onto the quotient spaces $X\/X_j$ and $X\/X_{jk}$\nrespectively (the Euclidean structure of $X$ allows us to identify\nthese abstract spaces with the subspaces of $X$ orthogonal to $X_j$\nand $X_{jk}$, but this is irrelevant here). Then $H$ may be written\nin the form\n\\begin{equation}\\label{eq:inonr}\n  H=h(P) + {\\textstyle\\sum_j v_j\\circ\\pi_j(x)+\n    \\sum_{j<k} v_{jk}\\circ\\pi_{jk}(x)}\n\\end{equation}  \nfor some real functions $v_j\\in\\cc_{\\mathrm{c}}(X\/X_j)$ and\n$v_{jk}\\in\\cc_{\\mathrm{c}}(X\/X_{jk})$. Thus Hamiltonians of the form\n\\eqref{eq:inonr} are natural objects in the $N$-body problem.  Note\nthat there should be no privileged origin in the momentum space, so\nif we accept $h(P)$ as an admissible kinetic energy operator then\n$h(P+p)$ should also be admissible for any\n$p=(p_1,\\dots,p_N)\\in(\\mathbb{R}^d)^N$.\n\nLet $\\mathscr{S}\\equiv\\mathscr{S}(X)$ be the set of linear subspaces of $X$ equipped\nwith the order relation $Y\\leq Z \\Leftrightarrow Y\\subset Z$. Then\n$\\mathscr{S}$ is a semilattice with $Y\\wedge Z =Y\\cap Z$.  If $Y\\in\\mathscr{S}$ then\nwe realize $\\cc_{\\mathrm{o}}(X\/Y)$ as a $C^*$-algebra of operators on $L^2(X)$ by\nassociating to $v$ the operator of multiplication by $v\\circ\\pi_Y$,\nwhere $\\pi_Y:X\\to X\/Y$ is the canonical surjection. The following\nfact is easy to prove: \\emph{if $\\mathcal{S}\\subset\\mathscr{S}$ is finite then\n$\\mathcal{C}(\\mathcal{S}):=\\sum_{Y\\in\\mathcal{S}} \\cc_{\\mathrm{o}}(X\/Y)$ is a direct topological sum and\nis an algebra if and only if $\\mathcal{S}$ is a sub-semilattice of $\\mathscr{S}$; in\nthis case, $\\mathcal{C}(\\mathcal{S})$ is an $\\mathcal{S}$-graded $C^*$-algebra}.\n\nIn the next proposition $\\mathcal{S}$ is the semilattice of subspaces of $X$\ngenerated by the $X_j$ and $X_{jk}$, i.e. the set of subspaces of\n$X$ obtained by taking arbitrary intersections of subspaces of the\nform $X_j$ and $X_{jk}$.  We denote $\\mathscr{T}_X$ the $C^*$-algebra of\noperators on $L^2(X)$ of the from $\\varphi(P)$ with\n$\\varphi\\in\\cc_{\\mathrm{o}}(X)$.\n\n\\begin{proposition}\\label{pr:mad}\n  Let $h:X\\to\\mathbb{R}$ be continuous with $h(x)\\to\\infty$ if\n  $x\\to\\infty$ and let $H_p$ be the self-adjoint operator\n  \\eqref{eq:inonr} with $h(P)$ replaced by $h(P+p)$. Then the\n  $C^*$-algebra generated by the operators $H_p$ when $p$ runs over\n  $(\\mathbb{R}^d)^N$ and $ v_j$ and $ v_{jk}$ run over the set of real\n  functions in $\\cc_{\\mathrm{c}}(X\/X_j)$ and $\\cc_{\\mathrm{c}}(X\/X_{jk})$ respectively is\n  $\\mathscr{C}=\\mathcal{C}(\\mathcal{S})\\cdot\\mathscr{T}_X=\\sum_{Y\\in\\mathcal{S}}\n  \\cc_{\\mathrm{o}}(X\/Y)\\cdot\\mathscr{T}_X$. Moreover, $\\mathscr{C}$ is $\\mathcal{S}$-graded by this\n  decomposition.\n\\end{proposition}\n\n \nThis has been proved in \\cite{DG1}. More general results of this\nnature are presented in Appendix \\ref{s:appb}.  Observe that we decided\nto fix the function $h$ which represents the kinetic energy but not\nthe potentials $v_j,v_{jk}$. However, as a consequence of\nProposition \\ref{pr:mad}, if we allow $h$ to vary we get the same\nalgebra.\n\nProposition \\ref{pr:mad} provides a basic example of ``Hamiltonian\nalgebra''.  We mention that $\\mathscr{C}$ is the crossed product of the\n$C^*$-algebra $\\mathcal{C}(\\mathcal{S})$ by the natural action of the additive group\n$X$, so it is a natural mathematical object.  We shall see in a more\ngeneral context that the set of self-adjoint operators affiliated to\nit is much larger than expected (cf. Theorem \\ref{th:iaff} for\nexample).\n\nIf we are in the nonrelativistic case and $V_j=0$ for all $j$ then\nthe center of mass of the system moves freely and it is more\nconvenient to eliminate it and to take as origin of the reference\nsystem the center of mass of the $N$ particles. Then the\nconfiguration space $X$ is the set of points\n$x=(x_1,\\dots,x_N)\\in(\\mathbb{R}^d)^N$ such that $\\sum_km_k x_k=0$.\nProposition \\ref{pr:mad}  remains valid if $\\mathcal{S}$ is conveniently\ndefined, see \\S \\ref{ss:cexample}.\n\nThe following ``generalized'' class of $N$-body systems is suggested\nby results from \\cite{Ma,Ma3}.\n\n\\begin{definition}\\label{df:nb}\n  An \\emph{$N$-body structure on a locally compact abelian group\n    $X$} is a set $\\mathcal{S}_X$ of closed subgroups such that\n  $X\\in\\mathcal{S}_X$ and such that for all $Y,Z\\in\\mathcal{S}_X$ the following\n  three conditions are satisfied: (i)~ $Y\\cap Z \\in \\mathcal{S}$; (ii) the\n  subgroups $Y,Z$ of $X$ are compatible; (iii) if $Y\\supsetneq Z$\n  then $Y\/Z$ is not compact.\n\\end{definition}\n$X$ must be thought as configuration space of the system. The notion\nof compatible subgroups is defined in Subsection \\ref{ss:compat} (if\n$X$ is a $\\sigma$-compact topological space this means that $Y+Z$ is\na closed subgroup).  We shall see that the Hamiltonian algebra\nassociated to such an $N$-body system is an interesting object:\n\\begin{equation}\\label{eq:gnbh}\n\\mathscr{C}_X(\\mathcal{S}_X):= \\mathcal{C}(\\mathcal{S}_X)\\cdot\\mathscr{T}_X \\cong \\mathcal{C}(\\mathcal{S}_X)\\rtimes X\n\\quad\\text{where}\\quad  \\mathcal{C}(\\mathcal{S}_X)=\n\\textstyle{\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}_X}} \\cc_{\\mathrm{o}}(X\/Y).\n\\end{equation} \n\nHere $\\mathscr{T}_X\\cong\\cc_{\\mathrm{o}}(X^*)$ is the group $C^*$-algebra of $X$ and\n$\\rtimes$ means crossed product.\n\n\\begin{example}\\label{ex:}\n  This framework covers an interesting extension of the standard\n  $N$-body setting. Assume that $X$ is a finite dimensional real\n  vector space.  In the standard framework the semilattice $\\mathcal{S}$\n  consists of linear subspaces of $X$ but here we allow them to be\n  closed additive subgroups. The closed additive\n  subgroups of $X$ are of the form $Y=E+L$ where $E$ is a vector\n  subspace of $X$ and $L$ is a lattice in a vector subspace $F$ of\n  $X$ such that $E\\cap F=\\{0\\}$.  More precisely, $L=\\sum_k\\mathbb{Z} f_k$\n  where $\\{f_k\\}$ is a basis in $F$. Thus $F\/L$ is a torus and if\n  $G$ is a third vector subspace such that $X=E\\oplus F\\oplus G$\n  then the space $X\/Y\\simeq (F\/L)\\oplus G$ is a cylinder with $F\/L$\n  as basis.\n\\end{example}\n\n\n\\subsection{The Euclidean many-body algebra}\n\\label{ss:sintro}\n\nWe introduce here an abstract framework which allows us to study\ncouplings between several $N$-body systems of the type considered\nabove. A concrete and physically interesting example may be found in\n\\S\\ref{ss:cexample}.\n\nLet $\\mathcal{X}$ be a real prehilbert space. Let $\\mathscr{S}(\\mathcal{X})$ be the set of\nfinite dimensional subspaces of $\\mathcal{X}$ equipped with the order\nrelation given by set inclusion. This is a lattice with $X\\wedge\nY=X\\cap Y$ and $X\\vee Y=X+Y$, but only the semilattice structure is\nrelevant for what follows. \n\n\nEach finite dimensional subspace\n$X\\subset \\mathcal{X}$ is equipped with the Euclidean structure induced by\n$\\mathcal{X}$ hence the Hilbert space $\\mathcal{H}_X= L^2(X)$ and the $C^*$-algebras\n$\\mathscr{L}_X=L(\\mathcal{H}_X)$ and $\\mathscr{K}_X=K(\\mathcal{H}_X)$ are well defined.  The group\nalgebra $\\mathscr{T}_X$ is defined as the closure in $\\mathscr{L}_X$ of the set of\noperators of convolution with functions of class $\\cc_{\\mathrm{c}}(X)$.  If\n$O=\\{0\\}$ then $\\mathcal{H}_O=\\mathbb{C}$ and $\\mathscr{L}_O=\\mathscr{K}_O=\\mathscr{T}_O=\\mathbb{C}$ by\nconvention. \n\nWe denote $h(P)$ the operator on $\\mathcal{H}_X$ given by\n$\\mathcal{F}_X^{-1}M_h\\mathcal{F}_X$, where $\\mathcal{F}_X$ is the Fourier transformation\nand $M_h$ is the operator of multiplication by the function\n$h:X\\to\\mathbb{C}$. Then $\\mathscr{T}_X=\\{\\psi(P)\\mid\\psi\\in\\cc_{\\mathrm{o}}(X)\\}$. We use the\nnotation $P=P_X$  if the space $X$ has to be specified. \n\nIf $X,Y$ are finite dimensional subspaces of $\\mathcal{X}$ we set\n$\\mathscr{L}_{XY}=L(\\mathcal{H}_Y,\\mathcal{H}_X)$ and $\\mathscr{K}_{XY}=K(\\mathcal{H}_Y,\\mathcal{H}_X)$.  We define\na closed subspace $\\mathscr{T}_{XY}\\subset\\mathscr{L}_{XY}$ as follows. If\n$\\varphi\\in\\cc_{\\mathrm{c}}(X+Y)$ then one may easily check that\n$(T_{XY}(\\varphi)f)(x)=\\int_Y\\varphi(x-y)f(y) \\text{d} y$ defines a\ncontinuous operator $\\mathcal{H}_Y\\to\\mathcal{H}_X$. Let\n\\begin{equation}\\label{eq:txy}\n\\mathscr{T}_{XY}= \\text{ norm closure of the set of operators} \\hspace{2mm}\nT_{XY}(\\varphi)  \\hspace{2mm} \\text{with} \\hspace{2mm} \n\\varphi\\in\\cc_{\\mathrm{c}}(X+Y).\n\\end{equation}\nClearly $\\mathscr{T}_{XX}=\\mathscr{T}_X$. The space $\\mathscr{T}_{XY}$ is a ``concrete''\nrealization of the Hilbert $C^*$-module introduced by Philip Green\nto show the Morita equivalence of the crossed products\n$\\cc_{\\mathrm{o}}(Z\/Y)\\rtimes X$ and $\\cc_{\\mathrm{o}}(Z\/X)\\rtimes Y$ where $Z=X+Y$ (this has\nbeen noticed by Georges Skandalis, see Remark \\ref{re:rief} for\nmore details).\n\n\nNow we fix a sub-semilattice $\\mathcal{S}\\subset\\mathscr{S}$, i.e. we assume that\n$X\\cap Y\\in\\mathcal{S}$ if $X,Y\\in\\mathcal{S}$.  This set completely determines the\nmany-body system and the class of Hamiltonians that we intend to\nstudy. For each $X\\in\\mathcal{S}$ the Hilbert space $\\mathcal{H}_X$ is thought as\nthe state space of an $N$-body system with $X$ as configuration\nspace.  We define the state space of the many-body system as the\nHilbertian direct sum\n\\begin{equation}\\label{eq:hs}\n\\mathcal{H}\\equiv\\mathcal{H}_\\mathcal{S}=\\oplus_{X\\in\\mathcal{S}} \\mathcal{H}_X.\n\\end{equation}\nWe have a natural embedding $\\mathscr{L}_{XY}\\subset L(\\mathcal{H})$ for all\n$X,Y\\in\\mathcal{S}$. Let $\\mathscr{L}\\equiv\\mathscr{L}_\\mathcal{S}$ be the closed linear span of the\nsubspaces $\\mathscr{L}_{XY}$. Clearly $\\mathscr{L}$ is a $C^*$-subalgebra of\n$L(\\mathcal{H})$ which is equal to $L(\\mathcal{H})$ if and only if $\\mathcal{S}$ is finite.\nWe will be interested in subspaces $\\mathscr{R}$ of $\\mathscr{L}$ constructed as\nfollows: for each couple $X,Y$ we are given a closed subspace\n$\\mathscr{R}_{XY}\\subset\\mathscr{L}_{XY}$ and $\\mathscr{R}\\equiv(\\mathscr{R}_{XY})_{X,Y\\in\\mathcal{S}}=\n{\\textstyle\\sum^\\mathrm{c}_{X,Y\\in\\mathcal{S}}}\\mathscr{R}_{XY}$ where $\\sum^\\mathrm{c}$ means\nclosure of the sum. Note that\n$\\mathscr{K}\\equiv\\mathscr{K}_\\mathcal{S}=(\\mathscr{K}_{XY})_{X,Y\\in\\mathcal{S}}=K(\\mathcal{H})$.\n\n\\begin{theorem}\\label{th:C}\n  Let $\\mathscr{T}\\equiv\\mathscr{T}_\\mathcal{S}=(\\mathscr{T}_{XY})_{X,Y\\in\\mathcal{S}}$. Then $\\mathscr{T}$ is a\n  closed self-adjoint subspace of $\\mathscr{L}$ and $\\mathscr{C}\\equiv\\mathscr{C}_\\mathcal{S}=\\mathscr{T}^2$\n  is a non-degenerate $C^*$-algebra of operators on $\\mathcal{H}$.\n\\end{theorem}\n\nWe say that \\emph{$\\mathscr{C}$ is the Hamiltonian algebra of the many-body\n  system $\\mathcal{S}$}.  This terminology will be justified later on: we\nshall see that physically interesting many-body Hamiltonians are\nself-adjoint operators affiliated to $\\mathscr{C}$. Moreover, in a quite\nprecise way, $\\mathscr{C}$ is the smallest $C^*$-algebra with this\nproperty. For the purposes of this paper \\emph{we define a\nmany-body Hamiltonian as a self-adjoint operator affiliated to\n$\\mathscr{C}$}.\n\nWe now equip $\\mathscr{C}$ with an $\\mathcal{S}$-graded $C^*$-algebras\nstructure. This structure will play a central role in the spectral\nanalysis of self-adjoint operators affiliated to $\\mathscr{C}$.  We often\nsay ``graded'' instead of $\\mathcal{S}$-graded.\n\nTo define the grading we need new objects.  If $Y\\not\\subset X$ we\nset $\\mathcal{C}_X(Y)=\\{0\\}$. If $Y\\subset X$ then we define $\\mathcal{C}_X(Y)$ as\nthe set of continuous functions on $X$ which are invariant under\ntranslations in the $Y$ directions and tend to zero in the $Y^\\perp$\ndirections.  This is a $C^*$-algebra of bounded uniformly continuous\nfunctions on $X$ canonically isomorphic with $\\cc_{\\mathrm{o}}(X\/Y)$ where $X\/Y$\nis the orthogonal of $Y$ in $X$. Thus\n\\begin{equation}\\label{eq:2cxy}\n\\mathcal{C}_X(Y)\\cong\\cc_{\\mathrm{o}}(X\/Y) \\text{ if } Y\\subset X \\quad\\text{and}\\quad\n\\mathcal{C}_X(Y)=\\{0\\} \\text{ if } Y\\not\\subset X.\n\\end{equation}\nLet $\\mathcal{C}_X\\equiv\\mathcal{C}_X(\\mathcal{S}):=\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}}\\mathcal{C}_X(Y)$, this is a\n$C^*$-algebra of bounded uniformly continuous functions on $X$.  We\nembed it in $\\mathscr{L}_X$ by identifying a function $\\varphi$ with the\noperator on $\\mathcal{H}_X$ of multiplication by $\\varphi$. Then\n\\begin{equation}\\label{eq:imp0}\n\\mathcal{C}\\equiv\\mathcal{C}_\\mathcal{S}=\\oplus_{X\\in\\mathcal{S}}\\mathcal{C}_X\n\\end{equation}\nis a $C^*$-algebra of operators on $\\mathcal{H}$ included in $\\mathscr{L}$.  For\neach $Z\\in\\mathcal{S}$ we define a $C^*$-subalgebra of $\\mathcal{C}$ by\n\\begin{equation}\\label{eq:impz}\n\\mathcal{C}(Z)\\equiv\\mathcal{C}_\\mathcal{S}(Z)=\\oplus_X\\mathcal{C}_X(Z)=\\oplus_{X\\supset Z}\\mathcal{C}_X(Z).\n\\end{equation}\nIt is easy to see that the family $\\{\\mathcal{C}(Z)\\}_{Z\\in\\mathcal{S}}$ defines a\ngraded $C^*$-algebra structure on $\\mathcal{C}$.\n\n\\begin{theorem}\\label{th:CG}\n  We have $\\mathscr{C}=\\mathscr{T}\\cdot\\mathcal{C}=\\mathcal{C}\\cdot\\mathscr{T}$.  For each $Z\\in\\mathcal{S}$ the\n  space $\\mathscr{C}(Z)=\\mathscr{T}\\cdot\\mathcal{C}(Z)=\\mathcal{C}(Z)\\cdot\\mathscr{T}$ is a $C^*$-subalgebra\n  of $\\mathscr{C}$ and the family $\\{\\mathscr{C}(Z)\\}_{Z\\in\\mathcal{S}}$ defines a graded\n  $C^*$-algebra structure on $\\mathscr{C}$.\n\\end{theorem}\n\nIn particular we get $\\mathscr{C}=(\\mathscr{C}_{XY})_{X,Y\\in\\mathcal{S}}$ with (the second\nand fourth equalities are not obvious):\n\\begin{align}\n\\mathscr{C}_{XY} \n&=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y \n={\\textstyle\\sum^\\mathrm{c}_{Z\\subset X\\cap Y}}\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Z)\n\\label{eq:icxy1} \\\\\n&=\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}= {\\textstyle\\sum^\\mathrm{c}_{Z\\subset X\\cap\n    Y}}\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}. \\label{eq:icxy2}\n\\end{align}\nThe $C^*$-algebras $\\mathcal{C}(Z)$ and $\\mathscr{C}(Z)$ ``live'' in the closed\nsubspace $\\mathcal{H}_{\\geq Z}=\\oplus_{X\\supset Z}\\mathcal{H}_X$ of $\\mathcal{H}$.  More\nprecisely, they leave invariant $\\mathcal{H}_{\\geq Z}$ and their restriction\nto its orthogonal subspace is zero. Moreover, if we denote\n$\\mathscr{L}_{\\geq Z}=(\\mathscr{L}_{X,Y})_{X,Y\\supset Z}\\subset L(\\mathcal{H}_{\\geq\n  Z})$ then clearly $\\mathcal{C}(Z)$ and $\\mathscr{C}(Z)$ are subalgebras of\n$\\mathscr{L}_{\\geq Z}$.\n\n\n\\begin{remark}\\label{re:nbody}{\\rm \nThe diagonal element $\\mathscr{C}_{XX}\\equiv\\mathscr{C}_X$ of the ``matrix'' $\\mathscr{C}$\nis given  by\n\\begin{equation}\\label{eq:NB}\n\\mathscr{C}_X=\\mathcal{C}_X\\cdot\\mathscr{T}_X=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\subset X}\\cc_{\\mathrm{o}}(X\/Z)\\cdot\\mathscr{T}_X}\n\\subset\\mathscr{L}_X.\n\\end{equation}\nThis $C^*$-algebra is \\emph{the Hamiltonian algebra of the\n  (generalized) $N$-body system} associated to the semi-lattice\n$\\mathcal{S}_X=\\{Z\\in\\mathcal{S}\\mid Z\\subset X\\}$ of subspaces of $X$,\ncf. \\eqref{eq:gnbh}. The non-diagonal elements $\\mathscr{C}_{XY}$ are\nHilbert $C^*$-bimodules which define the coupling between the\n$N$-body type systems $X$ and $Y$.  }\\end{remark}\n\n\\begin{remark}\\label{re:bigc}{\\rm Note that if we take $\\mathcal{S}$ equal\n    to the set of all finite dimensional subspaces of $\\mathcal{X}$ then we\n    get a graded $C^*$-algebra $\\mathscr{C}$ canonically associated to the\n    prehilbert space $\\mathcal{X}$. According to a remark in\n    \\S\\ref{ss:inotations}, any other choice of $\\mathcal{S}$ would give us a\n    graded $C^*$-subalgebra of this one.  }\\end{remark}\n\n\n\\begin{remark}\\label{re:sym}\n  The algebra $\\mathscr{C}$ is not adapted to symmetry considerations, in\n  particular in applications to physical systems consisting of\n  particles one has to assume them distinguishable. The Hamiltonian\n  algebra for systems of identical particles interacting through\n  field type forces (both bosonic and fermionic case) is constructed\n  in \\cite{Geo}. We mention that a quantum field model without\n  symmetry considerations corresponds to the case when $\\mathcal{S}$ is a\n  distributive relatively ortho-complemented lattice.\n\\end{remark}\n\n\n\\subsection{Particle systems with conserved total mass}\n\\label{ss:cexample} \n\n\nWe give now a physically interesting example of the preceding abstract\nconstruction.  We shall describe the many-body system associated to\n$N$ ``elementary particles'' of masses $m_1,\\dots,m_N$ moving in the\nphysical space $\\mathbb{R}^d$ without external fields. We shall get\n\\emph{a system in which the the total mass is conserved but not the\n  number of particles.}\n\nWe go back the framework of \\S \\ref{ss:inb} but assume that \nthe particles interact only through 2-body forces. Then\n\\begin{equation}\\label{eq:nonrel}\nH={\\textstyle\\sum_{j=1}^N P_j^2\/2m_j +\\sum_{j<k} V_{jk}(x_j-x_k)}.\n\\end{equation}\nand the center of mass of the system moves freely so it is\nconvenient to eliminate it. This is a standard procedure that we\nsketch now, cf.  \\cite{ABG,DeG1} for a detailed discussion of the\nformalism.  We take as origin of the reference system the center of\nmass of the $N$ particles so the configuration space $X$ is the set\nof points $x=(x_1,\\dots,x_N)\\in(\\mathbb{R}^d)^N$ such that $\\sum_km_k\nx_k=0$. We equip $X$ with the scalar product\n$\\braket{x}{y}=\\sum_{k=1}^N2m_k x_k y_k$. The advantage is that the\nreduced Hamiltonian, the operator acting in $L^2(X)$ naturally\nassociated to the expression \\eqref{eq:nonrel}, is\n$\\Delta_X+\\sum_{j<k}V_{jk}(x_j-x_k)$ where $\\Delta_X$ is the\nLaplacian associated to this scalar product. We denote by the same\nsymbol $H$ this reduced operator.\n\nThe first step is to describe the $C^*$-algebra generated by these\nHamiltonians, i.e. to get the analog of Proposition \\ref{pr:mad} in\nthe present context. Thus we have to describe the semilattice of\nsubspaces of $X$ generated by the $X_{(jk)}:=X_{jk}\\cap X$. We give\nthe result below and refer to the Appendix \\S\\ref{ss:anbody} for\nproofs.\n\nA partition $\\sigma$ of the set $\\{1,\\dots,N\\}$ is also called\ncluster decomposition. Then the sets of the partition are called\nclusters. A cluster $a\\in\\sigma$ is thought as a ``composite\nparticle'' of mass $m_a=\\sum_{k\\in a}m_k$.  Let $|\\sigma|$ be the\nnumber of clusters of $\\sigma$. We interpret $\\sigma$ as a system of\n$|\\sigma|$ particles with masses $m_a$ hence its configuration space\nshould be the set of $x=(x_a)_{a\\in\\sigma}\\in(\\mathbb{R}^d)^{|\\sigma|}$\nsuch that $\\sum_a m_a x_a=0$ equipped with a scalar product similar\nto that defined above.\n\nLet $X_\\sigma$ be the set of $x\\in X$ such that $x_i=x_j$ if $i,j$\nbelong to the same cluster and let us equip $X_\\sigma$ with the\nscalar product induced by $X$. Then there is an obvious isometric\nidentification of $X_\\sigma$ with the configuration space of the\nsystem $\\sigma$ as defined above. The advantage now is that all the\nspaces $X_\\sigma$ are isometrically embedded in the same $X$.  We\nequip the set $\\mathfrak{S}$ of partitions with the order relation:\n$\\sigma\\leq\\tau$ if and only if ``$\\tau$ is finer than $\\sigma$''\n(this is opposite to the usual convention). Then $\\sigma\\leq\\tau$ is\nequivalent to $X_\\sigma\\subset X_\\tau$ and $X_\\sigma\\cap\nX_\\tau=X_{\\sigma\\wedge\\tau}$.  Thus we see that $\\mathfrak{S}$ is\nisomorphic as semilattice with the set $\\mathcal{S}=\\{X_\\sigma\\mid\n\\sigma\\in\\mathfrak{S}\\}$ of subspaces of $X$ with inclusion as order\nrelation. Now it is easy to check that \\emph{$\\mathcal{S}$ coincides with\n  the semilattice of subspaces of $X$ generated by the $X_{(jk)}$}.\n\nWe abbreviate $\\mathcal{H}_\\sigma=\\mathcal{H}_{X_\\sigma}=L^2(X_\\sigma)$.  According\nto the identifications made above, this is the state space of a\nsystem of $|\\sigma|$ particles with masses $(m_a)_{a\\in\\sigma}$.\n\nNow we may apply our construction to $\\mathcal{S}$. We get a system whose\nstate space is $\\mathcal{H}=\\oplus_\\sigma \\mathcal{H}_{\\sigma}$. If the system is in\na state $u\\in\\mathcal{H}_{\\sigma}$ then it consists of $|\\sigma|$ particles\nof masses $m_a$. Note that $\\min\\mathfrak{S}$ is the partition\nconsisting of only one cluster $\\{1,\\dots,N\\}$ with mass\n$M=m_1+\\dots+m_N$. Since there are no external fields and we decided\nto eliminate the motion of the center of mass, this system must be\nthe vacuum. And its state space is indeed\n$\\mathcal{H}_{\\min\\mathfrak{S}}=\\mathbb{C}$. The algebra $\\mathscr{C}$ in this case\npredicts usual inter-cluster interactions associated, for examples,\nto potentials defined on $X^\\sigma=X\/X_\\sigma$, but also\ninteractions which force the system to make a transition from a\n``phase'' $\\sigma$ to a ``phase'' $\\tau$. In other terms, the system\nof $|\\sigma|$ particles with masses $(m_a)_{a\\in\\sigma}$ is\ntransformed into a system of $|\\tau|$ particles with masses\n$(m_b)_{b\\in\\tau}$. Thus the number of particles varies from $1$ to\n$N$ but the total mass is constant and equal to $M$.\n\n\n\n\n\\subsection{Natural morphisms and essential spectrum}\n\\label{ss:specan} \n\nWe return to the general case.  Sub-semilattices $\\mathcal{T}$ of $\\mathcal{S}$\ndefine many-body type subsystems (this is discussed in more detail\nin\\S\\ref{ss:subsys}). The spectral properties of the total many-body\nHamiltonian are described in terms of a special class of such\nsubsystems.\n\n\nEach $X\\in\\mathcal{S}$ determines a new many-body system $\\mathcal{S}_{\\geq\n  X}=\\{Y\\in\\mathcal{S}\\mid Y\\supset X\\}$ whose state space is $\\mathcal{H}_{\\geq\n  X}$. Let $\\mathscr{C}_{\\ge X}$ be the corresponding Hamiltonian algebra\n$\\mathscr{C}_{\\mathcal{S}_{\\geq X}}$. It is easy to see that $\\mathscr{C}_{\\geq\n  X}=\\sum^\\mathrm{c}_{Y\\supset X}\\mathscr{C}(Y)$.  Thus $\\mathscr{C}_{\\geq X}$ is a\n$C^*$-subalgebra of $\\mathscr{C}$ which lives and is non-degenerate on the\nsubspace $\\mathcal{H}_{\\geq X}$ of $\\mathcal{H}$. We mention one fact:\nif $\\Pi_{\\ge X}$ is the orthogonal projection $\\mathcal{H}\\to\\mathcal{H}_{\\ge X}$,\nthen $\\Pi_{\\ge X}\\mathscr{C}\\Pi_{\\ge X}$ is an $\\mathcal{S}$-graded\n$C^*$-subalgebra of $\\mathscr{C}_\\mathcal{S}$ and we have $\\mathscr{C}_{\\geq\n  X}\\subset\\Pi_{\\ge X}\\mathscr{C}\\Pi_{\\ge X}$ strictly in general.\n\n\nThen the general theory of graded\n$C^*$-algebras implies that \\emph{there is a unique linear\n  continuous projection $\\mathscr{P}_{\\geq X}:\\mathscr{C}\\to\\mathscr{C}_{\\geq X}$ such that\n  $\\mathscr{P}_{\\geq X}(T)=0$ if $T\\in\\mathscr{C}(Y)$ with $Y\\not\\supset X$ and this\n  projection is a morphism}. These are the \\emph{natural morphisms}\n  of the graded algebra $\\mathscr{C}$.\n\n\nThis extends to unbounded operators as follows: if $H$ is a\nself-adjoint operator on $\\mathcal{H}$ strictly affiliated to $\\mathscr{C}$ then\nthere is a unique self-adjoint operator $H_{\\ge X}\\equiv\\mathscr{P}_{\\ge\n  X}(H)$ on $\\mathcal{H}_{\\ge X}$ such that $\\mathscr{P}_{\\ge\n  X}(\\varphi(H))=\\varphi(H_{\\ge X})$ for all $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$.\nIf $H$ is only affiliated to $\\mathscr{C}$ then $H_{\\ge X}$ could be not\ndensely defined.\n\n\n\nAssume that the semilattice $\\mathcal{S}$ has a smallest element\n$\\min\\mathcal{S}$. Then $X\\in\\mathcal{S}$ is called \\emph{atom} if the only element\nof $\\mathcal{S}$ strictly included in $X$ is $\\min\\mathcal{S}$.  Let $\\mathcal{P}(\\mathcal{S})$ be\nthe set of atoms of $\\mathcal{S}$. We say that $\\mathcal{S}$ is \\emph{atomic} if\neach of its elements distinct from $\\min\\mathcal{S}$ contains an atom.  The\nfollowing HVZ type theorem is an immediate consequence Theorem\n\\ref{th:ga}. The symbol $\\overline{\\cup}$ means ``closure of\nunion''.\n\n\\begin{theorem}\\label{th:imp4}\nIf $H$ is a self-adjoint operator on $\\mathcal{H}$ strictly affiliated to\n$\\mathscr{C}$ then for each $X\\in\\mathcal{S}$ there is a unique self-adjoint\noperator $H_{\\geq X}\\equiv\\mathscr{P}_{\\geq X}(H)$ on $\\mathcal{H}_{\\geq X}$ such\nthat $\\mathscr{P}_{\\geq X} (\\varphi(H))=\\varphi(H_{\\geq X})$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$. The operator $H_{\\geq X}$ is strictly\naffiliated to $\\mathscr{C}_{\\geq X}$.  If $O\\in\\mathcal{S}$ and $\\mathcal{S}$ is atomic then\n\\begin{equation}\\label{eq:esi}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{X\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq X}).\n\\end{equation} \n\\end{theorem}\n\nThe theorem remains valid for operators which are only affiliated to\n$\\mathscr{C}$ but then we must allow them to be non-densely defined.\n\n\n\n\\subsection{Subsystems and subhamiltonians}\n\\label{ss:subsys}\n\n\n\nIf $\\mathcal{T}$ is an arbitrary subset of $\\mathcal{S}$ then the Hilbert space\n$\\mathcal{H}_\\mathcal{T}=\\oplus_{X\\in\\mathcal{T}}\\mathcal{H}_X$ is well defined and naturally\nembedded as a closed subspace of $\\mathcal{H}_\\mathcal{S}$. Let $\\Pi_\\mathcal{T}$ be the\northogonal projection of $\\mathcal{H}$ onto $\\mathcal{H}_\\mathcal{T}$. Note that the\ndefinition of $\\mathscr{T}_\\mathcal{S}$ makes sense for any set $\\mathcal{S}$ of finite\ndimensional subspaces , in particular we may replace $\\mathcal{S}$ by\n$\\mathcal{T}$. Then we have:\n\\begin{align}\n& \\mathscr{T}_\\mathcal{T} =\\Pi_\\mathcal{T}\\mathscr{T}_\\mathcal{S}\\Pi_\\mathcal{T}, \n\\quad\n\\mathcal{C}_\\mathcal{S}^\\mathcal{T} := \\Pi_\\mathcal{T}\\mathcal{C}_\\mathcal{S}\\Pi_\\mathcal{T}= \\oplus_{X\\in\\mathcal{T}}\\mathcal{C}_X(\\mathcal{S}), \n\\label{eq:subs1}\\\\\n& \\mathscr{C}_\\mathcal{S}^\\mathcal{T}:= \\Pi_\\mathcal{T}\\mathscr{C}_\\mathcal{S}\\Pi_\\mathcal{T}\n=\\Pi_\\mathcal{T}\\mathscr{T}_\\mathcal{S}\\cdot\\mathcal{C}_\\mathcal{S}\\Pi_\\mathcal{T}=\n\\mathscr{T}_\\mathcal{T}\\cdot\\mathcal{C}_\\mathcal{S}^\\mathcal{T} \\label{eq:subs2}\n\\end{align}\nFrom this we easily get that $\\mathscr{C}_\\mathcal{S}^\\mathcal{T}=\n{\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY}$ is an $\\mathcal{S}$-graded\n$C^*$-subalgebra of $\\mathscr{C}$ supported by the ideal generated by $\\mathcal{T}$\nin $\\mathcal{S}$ (an ideal is a subset $\\mathcal{J}$ of $\\mathcal{S}$ such that $X\\subset\nY\\in\\mathcal{J}\\Rightarrow X\\in\\mathcal{J}$).  The operators affiliated to\n$\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$ are affiliated to $\\mathscr{C}_\\mathcal{S}$, so are many-body\nHamiltonians in our sense. \n\nThe case when $\\mathcal{T}$ is a sub-semilattice of $\\mathcal{S}$ is interesting.\nIndeed, then $\\mathcal{T}$ defines a many-body system whose Hamiltonian\nalgebra $\\mathscr{C}_\\mathcal{T}$ is a $\\mathcal{T}$-graded $C^*$-algebra of operators on\nthe Hilbert space $\\mathcal{H}_\\mathcal{T}$. We emphasize that \\emph{this algebra\n  does not coincide with $\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$}.  We always have\n$\\mathscr{C}_\\mathcal{T}\\subset\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$ but \\emph{the inclusion is strict unless\n  $\\mathcal{T}$ is an ideal of $\\mathcal{S}$}.  This is clear because\n$\\mathcal{C}_\\mathcal{T}\\subset\\mathcal{C}_\\mathcal{S}^\\mathcal{T}$ strictly in general.\n\nThe simplest sub-semilattices are the chains (totally ordered\nsubsets). Then $\\mathcal{H}_\\mathcal{T}$ has a structure analogous to a Fock\nspace. Nonrelativistic Hamiltonians affiliated to such\n$C^*$-algebras $\\mathscr{C}_\\mathcal{S}^\\mathcal{T}$ have been studied before in \\cite{SSZ}.\nIf we take $\\mathcal{T}=\\{X\\}$ for an arbitrary $X\\in\\mathcal{S}$ then the\nassociated subsystem has $\\mathcal{H}_X$ as state space and its Hamiltonian\nalgebra is just $\\mathscr{C}_X=\\mathcal{C}_X(\\mathcal{S})\\cdot\\mathscr{T}_X$, the Hamiltonian\nalgebra of the (generalized) $N$-body system determined by the\nsemi-lattice $\\mathcal{S}_X$. We refer to \\S\\ref{ss:T} and to the Example\n\\ref{ex:fried} for other simple but instructive examples of\nsubsystems.\n\n\n\n\\subsection{Intrinsic descriptions}\n\\label{ss:id}\n\nWe give two explicit descriptions of $\\mathscr{C}(Z)$ as an algebra of\noperators on $\\mathcal{H}_{\\geq Z}$. Since $\\mathscr{C}$ is the closure of the sum\nof the algebras $\\mathscr{C}(Z)$, these descriptions allow one to check\nrather easily whether a self-adjoint operator is affiliated to $\\mathscr{C}$\nor not. Both theorems are consequences of more general results in\nSection \\ref{s:id}.\n\nFor any vector $a\\in\\mathcal{X}$ and any finite dimensional subspace $X$ of\n$\\mathcal{X}$ we define two unitary operators in $\\mathcal{H}_X$ by\n$(U_af)(x)=f(x+a_X)$ and $(V_a f)(x)=\\mathrm{e}^{i\\braket{a}{x}} f(x)$\nwhere $a_X$ is the orthogonal projection of $x$ on $X$. Then\n$\\{U_a\\}_{a\\in\\mathcal{X}}$ and $\\{V_a\\}_{a\\in\\mathcal{X}}$ are strongly continuous\nrepresentations of (the additive group) $\\mathcal{X}$ on $\\mathcal{H}_X$ such that\n$U_a=1 \\Leftrightarrow V_a=1 \\Leftrightarrow a\\perp X$. The direct\nsum over $X\\in\\mathcal{S}$ of these representations give representations of\n$\\mathcal{X}$ on $\\mathcal{H}$ for which we use the same notations.\n\n\n\\begin{theorem}\\label{th:CZU}\n  $\\mathscr{C}(Z)$ is the set of $T\\in\\mathscr{L}_{\\geq Z}$ such that\n\\begin{compactenum}\n\\item[{\\rm(i)}] $U_a^*T U_a=T$ for all $a\\in Z$ and $\\|T(V_a-1)\\|\\to\n  0$ if $a\\to 0$ in $Z^\\perp$ ,\n\\item[{\\rm(ii)}] $\\|T(U_a-1)\\|\\to 0$ and $\\|V^*_a T V_a-T\\|\\to 0$ if\n  $a\\to 0$ in $\\mathcal{X}$.\n\\end{compactenum}\n\\end{theorem}\n\n\nLet $\\mathcal{S}\/Z$ be the set of subspaces of $\\mathcal{X}\/Z=Z^\\perp$ of the form\n$X\/Z$ with $X\\in\\mathcal{S}, X\\supset Z$. Clearly $\\mathcal{S}\/Z$ is a semilattice\nof finite dimensional subspaces of $\\mathcal{X}\/Z$ so the Hilbert space\n$\\mathcal{H}_{\\mathcal{S}\/Z}$ and the corresponding algebra of compact operators\n$\\mathscr{K}_{\\mathcal{S}\/Z}$ are well defined.  If $X\\supset Z$ then $X=Z\\oplus\nX\/Z$ so we have a canonical factorization\n$\\mathcal{H}_X=\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}$. Thus $\\mathcal{H}_{\\geq\n  Z}=\\mathcal{H}_Z\\otimes\\mathcal{H}_{\\mathcal{S}\/Z}$.\n\n\\begin{theorem}\\label{th:CZ}\n$\\mathscr{C}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{\\mathcal{S}\/Z}$ relatively to the factorization\n$\\mathcal{H}_{\\geq Z}=\\mathcal{H}_Z\\otimes\\mathcal{H}_{\\mathcal{S}\/Z}$.\n\\end{theorem}\n\n\\subsection{Factorization properties}\n\\label{ss:fact}\n\nFor $Z\\subset X\\cap Y$ we have $X=Z\\oplus (X\/Z)$ and $Y=Y\\oplus\n(Y\/Z)$ hence we have canonical factorizations\n\\begin{equation}\\label{eq:xyzint}\n\\mathcal{H}_X=\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}\n\\quad\\text{and}\\quad \n\\mathcal{H}_Y=\\mathcal{H}_Z\\otimes\\mathcal{H}_{Y\/Z}.\n\\end{equation}\nRelatively to these factorizations, we get from Theorem \\ref{th:CZ}:\n\\begin{equation}\\label{eq:CZ}\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{X\/Z,Y\/Z} \\cong \\cc_{\\mathrm{o}}(Z^*;\\mathscr{K}_{X\/Z,Y\/Z}).\n\\end{equation}\nThe tensor product (and those below) is in the category of Hilbert\nmodules, cf.  \\S\\ref{ss:ha}. We have written $Z^*$ above in spite of\nthe canonical Euclidean isomorphism $Z^*\\cong Z$ in order to stress\nthat we consider functions of momentum not of position. For any\n$X,Y$ we set\n\\begin{equation}\\label{eq:xony}\nX\/Y=X\/(X\\cap Y)= X\\ominus (X\\cap Y)\n\\end{equation}\nand so we have \n\\begin{equation}\\label{eq:xyonz}\nX\/Z=X\/(X\\cap Y)\\oplus (X\\cap Y)\/Z = X\/Y\\oplus (X\\cap Y)\/Z \n\\end{equation}\nand similarly for $Y\/Z$. Then from \\eqref{eq:CZ} and\n\\eqref{eq:comtens} we get the finer factorization:\n\\begin{equation}\\label{eq:xyzeptens}\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{(X\\cap Y)\/Z}\n\\otimes \\mathscr{K}_{X\/Y,Y\/X}. \n\\end{equation}\nIn particular, we get\n\\begin{equation}\\label{eq:xyzeint}\n\\mathscr{C}_{XY}=\\mathscr{C}_{X\\cap Y}\\otimes \\mathscr{K}_{X\/Y,Y\/X}\n\\end{equation}\nrelatively to the tensor factorizations \n\\begin{equation}\\label{eq:xyeht}\n\\mathcal{H}_X=\\mathcal{H}_{X\\cap Y}\\otimes\\mathcal{H}_{X\/Y} \\quad\\text{and}\\quad \n\\mathcal{H}_Y=\\mathcal{H}_{X\\cap Y}\\otimes\\mathcal{H}_{Y\/X}.\n\\end{equation}\nSince $\\mathscr{K}_{X\/Y,O}\\cong\\mathcal{H}_{X\/Y}$\nin the special case $Z\\subset Y\\subset X$ we have \n\\begin{equation}\\label{eq:xby}\n\\mathscr{C}_{XY}=\\mathscr{C}_{Y}\\otimes\\mathcal{H}_{X\/Y}\n\\quad\\text{and}\\quad\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{Y\/Z} \\otimes \\mathcal{H}_{X\/Y}.\n\\end{equation}\n\n\n\\begin{example}\\label{ex:fried}\n  These factorizations give us the possibility of expressing quite\n  explicitly the Hamiltonian algebra of some subsystems. We refer to\n  \\S\\ref{ss:T} for more general situations and consider here\n  sub-semilattices of the form $\\mathcal{T}=\\{X,Y\\}$ with $X\\supset Y$.\n  This is a toy model, an $N$-body system coupled to one of its\n  subsystems, and can be nicely formulated in a purely abstract\n  setting, cf. Proposition \\ref{pr:toymodel}.  We have\n  $\\mathcal{H}_\\mathcal{T}=\\mathcal{H}_X\\oplus\\mathcal{H}_Y$ with $\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}$.\n  From \\eqref{eq:xby} we have\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}=\n\\begin{pmatrix}\n\\mathscr{C}_X & \\mathscr{C}_{Y}\\otimes\\mathcal{H}_{X\/Y}\\\\\n\\mathscr{C}_{Y}\\otimes\\mathcal{H}^*_{X\/Y} & \\mathscr{C}_Y\n\\end{pmatrix}.\n\\]\nwhere $\\mathcal{H}^*_{X\/Y}$ has a natural meaning (see\n\\S\\ref{ss:affil}). The grading is defined for $Z\\in\\mathcal{S}_X$\nby\n\\begin{compactenum}\n\\item\nIf  $Z\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(Z)=\n\\begin{pmatrix}\n\\mathscr{C}_X(Z) & \\mathscr{C}_{Y}(Z)\\otimes\\mathcal{H}_{X\/Y}\\\\\n\\mathscr{C}_{Y}(Z)\\otimes\\mathcal{H}^*_{X\/Y} & \\mathscr{C}_Y(Z)\n\\end{pmatrix}.\n\\]\n\\item   \\label{p:i2ex}\nIf  $Z\\not\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(Z)=\n\\begin{pmatrix}\n\\mathscr{C}_X(Z) & 0\\\\\n0 & 0\n\\end{pmatrix}.\n\\]\n\\end{compactenum}\nIf $\\mathcal{T}=\\{X,O\\}$ we get a version of the Friedrichs model: an\n$N$-body system coupled to the vacuum. The case when $\\mathcal{T}$ is an\narbitrary chain (a totally ordered subset of $\\mathcal{S}$) is very similar.\nThe case $\\mathcal{T}=\\{X,Y\\}$ with not comparable $X,Y$ is more complicated\nand is treated in \\S\\ref{ss:T} in a more general setting.\n\\end{example}\n\n\n\n\\subsection{Examples of many-body Hamiltonians}\n\\label{ss:ex}\n\n\nHere we use Theorems \\ref{th:CZ} and \\ref{th:CZU} to construct\nself-adjoint operators strictly affiliated to $\\mathscr{C}$. For simplicity,\n\\emph{in this and the next subsections $\\mathcal{S}$ is assumed finite}. If\n$\\mathcal{S}$ is infinite then an assumption of the same nature as the\nnon-zero mass condition in quantum field theory models is needed to\nensure that the kinetic energy operator $K$ is affiliated to $\\mathscr{C}$.\n\nThe Hamiltonians will be of the form $H=K+I$ where the self-adjoint\noperator $K$ is the kinetic energy and $I$ is an\ninteraction term bounded in form sense by $K$. More precisely, $I$\nis a symmetric sesquilinear form on the domain of $|K|^{1\/2}$ which\nis continuous, i.e. satisfies\n\\begin{equation}\\label{eq:formbound}\n\\pm I\\leq \\mu|K+ia| \\hspace{2mm}\\text{for some real numbers } \\mu, a.\n\\end{equation}\n$H$ and $K$ are matrices of operators, e.g. $H=(H_{XY})_{X,Y\\in\\mathcal{S}}$\nwhere $H_{XY}$ is defined on a subspace of $\\mathcal{H}_Y$ and has values in\n$\\mathcal{H}_X$ and the relation $H_{XY}^*=H_{YX}$ holds at least\nformally. By construction $K$ is given by a diagonal matrix, so\n$K_{XY}=0$ if $X\\neq Y$, and we set $K_X=K_{XX}$. The interaction\nwill be a matrix of sesquilinear forms. Then $H_{XX}=K_{X}+I_{XX}$\nwill be an $N$-body type Hamiltonian, i.e. a self-adjoint operator\naffiliated to $\\mathscr{C}_X$, cf. Remark \\ref{re:nbody}. The non-diagonal\nelements $H_{XY}=I_{XY}$ define the interaction between the systems\n$X$ and $Y$.  We give now a rigorous construction of such\nHamiltonians.\n\n\n{\\bf(a)} For each $X$ we choose a kinetic energy operator\n$K_X=h_X(P)$ for the system having $X$ as configuration space. The\nfunction $h_X:X\\to\\mathbb{R}$ is continuous and such that\n$h_X(x)\\to\\infty$ if $x\\to\\infty$.  We stress that there are no\nrelations between the kinetic energies of the systems corresponding\nto different $X$. Denote $\\mathcal{G}^2_X$ the domain of $K_X$ equipped with\nthe graph norm and let $\\mathcal{G}^s_X$ ($s\\in\\mathbb{R}$) be the scale of\nHilbert spaces associated to it, e.g.  $\\mathcal{G}^0_X=\\mathcal{H}_X$,\n$\\mathcal{G}^1_X=D(K_X^{1\/2})$ is the form domain of $K_X$, and $\\mathcal{G}^{-1}_X$\nits adjoint space.\n\n\n{\\bf(b)} The total kinetic energy of the system is by definition\n$K=\\oplus_X K_X$. We call this a \\emph{standard kinetic energy\n  operator}.  Then the spaces $\\mathcal{G}^s$ of the scale determined by the\ndomain $\\mathcal{G}^2$ of $K$ can be identified with direct sums\n$\\mathcal{G}^s\\simeq\\oplus_X\\mathcal{G}^s_X$. In particular this holds for the form\ndomain $\\mathcal{G}^1=D(K^{1\/2})$ and for its adjoint space $\\mathcal{G}^{-1}$.\nNote that we may also introduce the operators $K_{\\geq\n  X}=\\oplus_{Y\\supset X} K_Y$ and the associated spaces $\\mathcal{G}^s_{\\ge\n  X}$. If $s>0$ we have $\\mathcal{G}^s_{\\ge X}=\\mathcal{G}^s\\cap\\mathcal{H}_{\\ge X}$.\n\n{\\bf(c)} The simplest type of interactions that we may consider are\ngiven by symmetric elements $I$ of the multiplier algebra of\n$\\mathscr{C}$. Then $H=K+I$ is strictly affiliated to $\\mathscr{C}$ and $\\mathscr{P}_{\\geq\n  X}(H)=K_{\\geq X}+\\mathscr{P}_{\\geq X} (I)$ where $\\mathscr{P}_{\\geq X}$ is\nextended to the multiplier algebras as explained in\n\\cite[p. 18]{La}.\n\n{\\bf(d)} In order to cover singular interactions (form bounded but\nnot necessarily operator bounded by $K$) we assume that the\nfunctions $h_X$ are equivalent to regular weights. This is a quite\nweak assumption, cf. page \\pageref{p:regw}.  For example, it\nsuffices that $c'|x|^{\\alpha}\\leq h_X(x)\\leq c''|x|^{\\alpha}$ for\nlarge $x$ where $c',c'',\\alpha>0$ are numbers depending on $X$.\nThen $U_a,V_a$ induce continuous operators in each of the spaces\n$\\mathcal{G}^s_X$, $\\mathcal{G}^s$, $\\mathcal{G}^s_{\\ge X}$. \n\n\n{\\bf(e)} The interaction will be of the form $I=\\sum_{Z\\in\\mathcal{S}} I(Z)$\nwhere the $I(Z)$ are continuous symmetric sesquilinear forms on\n$\\mathcal{G}^1$ such that $ I(Z) \\geq -\\mu_Z K -\\nu$ for some positive\nnumbers $\\mu_Z$ and $\\nu$ with $\\sum_Z\\mu_Z<1$. Then the form sum\n$K+I$ defines a self-adjoint operator $H$ on $\\mathcal{H}$.\n\n{\\bf(f)} We identify $I(Z)$ with a symmetric operator\n$\\mathcal{G}^1\\to\\mathcal{G}^{-1}$ and we assume that $I(Z)$ is supported by the\nsubspace $\\mathcal{H}_{\\ge Z}$.  In other terms, $I(Z)$ is the sesquilinear\nform on $\\mathcal{G}^1$ associated to an operator $I(Z):\\mathcal{G}^1_{\\ge\n  Z}\\to\\mathcal{G}^{-1}_{\\ge Z}$.  Moreover, we assume that this last\noperator satisfies\n\\begin{equation}\\label{eq:iaff}\nU_a I(Z)=I(Z) U_a \\text{ if } a\\in Z, \\ \nI(Z)(V_a-1)\\to 0 \\text{ if } a\\to 0 \\text{ in } Z^\\perp, \\ \nV^*_a I(Z) V_a\\to I(Z) \\text{ if } a\\to 0\n\\end{equation}\nwhere the limits hold in norm in $L(\\mathcal{G}^2_{\\ge Z},\\mathcal{G}^{-1}_{\\ge Z})$.\n\nNote that the first part of condition {\\bf(f)}, saying that $I(Z)$\nis supported by $\\mathcal{H}_{\\ge Z}$, is equivalent to an estimate of the\nform $\\pm I(Z)\\le\\mu K_{\\geq Z}+ \\nu\\Pi_{\\geq Z}$ for some positive\nnumbers $\\mu,\\nu$. See also Remark \\ref{re:iaff}.\n\n\\begin{theorem}\\label{th:iaff}\n  The Hamiltonian $H$ is a self-adjoint operator strictly affiliated\n  to $\\mathscr{C}$, we have   \n  $H_{\\geq X}=K_{\\geq X}+\\sum_{Z\\geq X} I(Z)$, \n  and $\\mathrm{Sp_{ess}}(H)=\\textstyle{\\bigcup}_{X\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq X})$.\n\\end{theorem}\n\n\\begin{remark}\\label{re:pauli}\n  We required the $h_X$ to be bounded from below only for the\n  simplicity of the statements. Moreover, a simple extension of the\n  formalism allows one to treat particles with arbitrary\n  spin. Indeed, if $E$ is a complex Hilbert then Theorem \\ref{th:CG}\n  remains true if $\\mathscr{C}$ is replaced by $\\mathscr{C}^E=\\mathscr{C}\\otimes K(E)$ and\n  the $\\mathscr{C}(Z)$ by $\\mathscr{C}(Z)\\otimes K(E)$. If $E$ is the spin space\n  then it is finite dimensional and one obtains $\\mathscr{C}^E$ exactly as\n  above by replacing the $\\mathcal{H}(X)$ by $\\mathcal{H}(X)\\otimes\n  E=L^2(X;E)$. Then one may consider instead of scalar kinetic\n  energy functions $h$ self-adjoint operator valued functions\n  $h:X^*\\to L(E)$. For example, we may take as one particle kinetic\n  energy operators the Pauli or Dirac Hamiltonians.\n\\end{remark}\n\n\n\\begin{remark}\\label{re:iaff}\nWe give here a second, more explicit version of condition\n{\\bf(f)}. Since \n$I(Z)$ is a continuous symmetric operator $\\mathcal{G}^1\\to\\mathcal{G}^{-1}$ we may\nrepresent it as a matrix $I(Z)=(I_{XY}(Z))_{X,Y\\in\\mathcal{S}}$ of\ncontinuous operators $I_{XY}(Z):\\mathcal{G}^1_Y\\to\\mathcal{G}^{-1}_X$ with\n$I_{XY}(Z)^*=I_{YX}(Z)$. We take $I_{XY}(Z)=0$ if $Z\\not\\subset\nX\\cap Y$ and if $Z\\subset X\\cap Y$ we assume\n$V^*_a I_{XY}(Z) V_a\\to I_{XY}(Z) \\text{ if } a\\to 0 \\text{ in }\nX+Y$ and \n\\begin{equation}\\label{eq:iiaff}\nU_a I_{XY}(Z)=I_{XY}(Z) U_a \\text{ if }a\\in Z,  \\quad\nI_{XY}(Z)(V_a-1)\\to 0 \\text{ if } a\\to 0 \\text{ in } Y\/Z.\n\\end{equation}\nThe limits should hold in norm in $L(\\mathcal{G}^2_Y,\\mathcal{G}^{-1}_X)$.\n\\end{remark}\n\nThe operators $I_{XY}(Z)$satisfying \\eqref{eq:iiaff} are described\nin more detail in Proposition \\ref{pr:zxy}. In the next example we\nconsider the simplest situation which is useful in the\nnonrelativistic case.\n\nIf $E$ is an Euclidean space and $s$ is a real number let $\\mathcal{H}^s_E$\nbe the Sobolev space defined by the norm\n\\[\n\\|u\\|_{\\mathcal{H}^s} = \\|(1+\\Delta_E)^{s\/2}u\\|\n\\] \nwhere $\\Delta_E$ is the (positive) Laplacian associated to the\nEuclidean space $E$. The space $\\mathcal{H}^s_E$ is equipped with two\ncontinuous representations of $E$, a unitary one induced by\n$\\{U_x\\}_{x\\in E}$ and a non-unitary one induced by $\\{V_x\\}_{x\\in\n  E}$.  If $E=O:=\\{0\\}$ we define $\\mathcal{H}_E^s=\\mathbb{C}$.\n\n\\begin{definition}\\label{df:small}\n  If $E,F$ are Euclidean spaces and $T:\\mathcal{H}^s_E\\to\\mathcal{H}^t_F$ is a\n  linear map, we say that \\emph{$T$ is small at infinity} if there\n  is $\\varepsilon>0$ such that when viewed as a map\n  $\\mathcal{H}^{s+\\varepsilon}_E\\to\\mathcal{H}^{t}_F$ the operator $T$ is compact.\n\\end{definition}\nBy the closed graph theorem $T$ is continuous and the compactness\nproperty holds for all $\\varepsilon>0$.  If $E=O$ or $F=O$ then we\nconsider that all the operators $T:\\mathcal{H}^s_E\\to\\mathcal{H}^t_F$ are small at\ninfinity.\n\n\\begin{example}\\label{ex:zxy}\n  Due to assumption {\\bf(d)} the form domains of $K_X$ and $K_Y$ are\n  Sobolev spaces, for example $\\mathcal{G}^1_X=\\mathcal{H}^s_X$ and\n  $\\mathcal{G}^1_Y=\\mathcal{H}^t_Y$. Let $I_{XY}^Z:\\mathcal{H}^t_{Y\/Z}\\to\\mathcal{H}^{-s}_{X\/Z}$ be\n  a linear small at infinity map. Then we may take\n  $I_{XY}(Z)=1_Z\\otimes I_{XY}^Z$ relatively to the tensor\n  factorizations \\eqref{eq:xyzint}.\n\\end{example}\n\n\nWe make now some comments to clarify the conditions {\\bf(a)} -\n{\\bf(f)}.  Assume, more generally, that $\\mathscr{C}$ is a $C^*$-algebra of\noperators on a Hilbert space $\\mathcal{H}$ and that $K$ is a self-adjoint\noperator on $\\mathcal{H}$ affiliated to $\\mathscr{C}$. Let $I$ be a continuous\nsymmetric sesquilinear form on the domain of $|K|^{1\/2}$. Then for\nsmall real $\\nu$ the form sum $K+\\nu I$ is a self-adjoint operator\n$H_\\nu$. If $H_\\nu$ is affiliated to $\\mathscr{C}$ for small $\\nu$, and\nsince the derivative with respect to $\\nu$ at zero of\n$(H_\\nu+i)^{-1}$ exists in norm, we get\n$(K+i)^{-1}I(K+i)^{-1}\\in\\mathscr{C}$. This clearly implies\n$\\jap{K}^{-2}I\\jap{K}^{-2}\\in\\mathscr{C}$.  Since\n$\\jap{K}^{-1\/2}I\\jap{K}^{-1\/2}$ is a bounded operator, the map\n$z\\mapsto\\jap{K}^{-z}I\\jap{K}^{-z}$ is holomorphic on $\\Re{z}>1\/2$\nhence we get\n\\begin{equation}\\label{eq:ent}\n\\jap{K}^{-\\alpha}I\\jap{K}^{-\\alpha}\\in\\mathscr{C} \\text{ if } \\alpha>1\/2. \n\\end{equation}\nReciprocally, if $K$ is strictly affiliated to $\\mathscr{C}$ (and $K$ as\ndefined at (b) has this property) then Theorem 2.8 from \\cite{DG3}\nsays that $\\jap{K}^{-1\/2}I\\jap{K}^{-\\alpha}\\in\\mathscr{C}$ suffices to\nensure that $H=K+I$ is strictly affiliated to $\\mathscr{C}$ under a quite\ngeneral condition needed to make this operator well defined (this is\nthe role of assumption (e) above).  Condition (f) is formulated such\nas to imply $\\jap{K}^{-1\/2}I\\jap{K}^{-1}\\in\\mathscr{C}$. To simplify the\nstatement we added condition (d) which implies that the spaces\n$\\mathcal{G}^s$ are stable under the group $V_a$. Formally\n\\[\n(\\jap{K}^{-1\/2}I\\jap{K}^{-1})_{XY}=\n\\jap{K_X}^{-1\/2}I_{XY}\\jap{K_Y}^{-1}.\n\\]\nSo this should belong to $\\mathscr{C}_{XY}=\\sum_{Z\\subset X\\cap\n  Y}\\mathscr{C}_{XY}(Z)$. Thus $I_{XY}$ must be a sum of terms $I_{XY}(Z)$\nwith\n\\[\n\\jap{K_X}^{-1\/2}I_{XY}(Z)\\jap{K_Y}^{-1}\\in\\mathscr{C}_{XY}(Z). \n\\]\nConditions (d) and (f) are formulated such as this to hold,\ncf. Remark \\ref{re:iaff} and Theorem \\ref{th:CZU}.\n\n\n\n\\subsection{Pauli-Fierz   Hamiltonians}\n\\label{ss:affil} \n\nThe next result is an a priori argument which supports our\ninterpretation of $\\mathscr{C}$ as Hamiltonian algebra of a many-body\nsystem: we show that $\\mathscr{C}$ is the $C^*$-algebra generated by a\nsimple class of Hamiltonians which have a natural quantum field\ntheoretic interpretation. For simplicity we state this only for\nfinite $\\mathcal{S}$.\n\nFor each couple $X,Y\\in\\mathcal{S}$ such that $X\\supset Y$ we have\n$\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}$.  Then we define\n$\\Phi_{XY}\\subset\\mathscr{L}_{XY}$ as the closed linear subspace consisting\nof ``creation operators'' associated to states from $\\mathcal{H}_{X\/Y}$,\ni.e. operators $a^*(\\theta):\\mathcal{H}_Y\\to\\mathcal{H}_X$ with $\\theta\\in\\mathcal{H}_{X\/Y}$\nwhich act as $u\\mapsto u\\otimes\\theta$.  We set\n$\\Phi_{YX}=\\Phi_{XY}^*\\subset\\mathscr{L}_{YX}$, this is the space of\n``annihilation operators'' $a(\\theta)=a^*(\\theta)^*$ defined by\n$\\mathcal{H}_{X\/Y}$.  This defines $\\Phi_{XY}$ when $X,Y$ are comparable,\ni.e. $X\\supset Y$ or $X\\subset Y$, which we abbreviate by $X\\sim\nY$. If $X\\not\\sim Y$ then we take $\\Phi_{XY}=0$. Note that\n$\\Phi_{XX}=\\mathbb{C} 1_X$, where $1_X$ is the identity operator on\n$\\mathcal{H}_X$.  We have\n\\begin{equation}\\label{eq:Phi}\n\\mathscr{T}_X\\cdot\\Phi_{XY}=\\Phi_{XY}\\cdot \\mathscr{T}_Y=\\mathscr{T}_{XY} \\quad\n\\text{if } X\\sim Y.\n\\end{equation}\n\nNow let $\\Phi=(\\Phi_{XY})_{X,Y\\in\\mathcal{S}}\\subset\\mathscr{L}$. This is\na closed self-adjoint linear space of bounded operators on $\\mathcal{H}$.  A\nsymmetric element $\\phi\\in\\Phi$ will be called \\emph{field\n  operator}. Giving such a $\\phi$ is equivalent to giving a family\n$\\theta=(\\theta_{XY})_{X\\supset Y}$ of elements\n$\\theta_{XY}\\in\\mathcal{H}_{X\/Y}$, the components of the operator\n$\\phi\\equiv\\phi(\\theta)$ being given by:\n$\\phi_{XY}=a^*(\\theta_{XY})$ if $X\\supset Y$,\n$\\phi_{XY}=a(\\theta_{YX})$ if $X\\subset Y$, and $\\phi_{XY}=0$ if\n$X\\not\\sim Y$. \n\nThe operators of the form $K+\\phi$, where $K$ is a standard kinetic\nenergy operator and $\\phi\\in\\Phi$ is a field operator, will be\ncalled \\emph{Pauli-Fierz Hamiltonians}.\n\n\\begin{theorem}\\label{th:motiv}\n  If $\\mathcal{S}$ is finite then $\\mathscr{C}$ is the $C^*$-algebra\n  generated by the Pauli-Fierz Hamiltonians.\n\\end{theorem}\n\nThus $\\mathscr{C}$ is generated by a class of Hamiltonians involving only\nelementary field type interactions. On the other hand, we have seen\nbefore that the class of Hamiltonians affiliated to $\\mathscr{C}$ is very\nlarge and covers \\mbox{$N$-body} systems interacting between\nthemselves with field type interactions. We emphasize that the\n$k$-body type interactions \\emph{inside} each of the $N$-body\nsubsystems are generated by pure field interactions.\n\n\n\\subsection{Nonrelativistic Hamiltonians and Mourre\n  estimate}\n\\label{ss:mouint}  \n\nWe prove the Mourre estimate only for nonrelativistic many-body systems. \nThere are serious difficulties when\nthe kinetic energy is not a quadratic form even in the much simpler\ncase of $N$-body Hamiltonians, but see \\cite{De1,Ger1,DG2} for some\npartial results which could be extended to our setting. Note that\nthe quantum field case is much easier from this point of view\nbecause of the special nature of the interactions\n\\cite{DeG2,Ger2,Geo}. \n\nLet $\\mathcal{S}$ be a finite semilattice of subspaces of $\\mathcal{X}$.  Recall\nthat for $X\\in\\mathcal{S}$ we denote $\\mathcal{S}\/X$ the set of subspaces $Y\/X=Y\\cap\nX^\\perp$ with $Y\\in\\mathcal{S}_{\\ge X}$. This is a finite semilattice of\nsubspaces of $\\mathcal{X}$ which contains $O$. Hence the Hilbert space\n$\\mathcal{H}_{\\mathcal{S}\/X}$ and the $C^*$-algebra $\\mathscr{C}_{\\mathcal{S}\/X}$ are well defined\nby our general rules and (cf. \\S\\ref{ss:fact}):\n\\begin{equation}\\label{eq:f}\n\\mathcal{H}_{\\geq X}=\\mathcal{H}_X\\otimes\\mathcal{H}_{\\mathcal{S}\/X}\\quad \\text{and} \\quad\n\\mathscr{C}_{\\geq X}=\\mathscr{T}_X\\otimes\\mathscr{C}_{\\mathcal{S}\/X} .\n\\end{equation}\nDenote $\\Delta_X$ the (positive) Laplacian associated to the Euclidean\nspace $X$ with the convention $\\Delta_O=0$. We have\n$\\Delta_X=h_X(P)$ with $h_X(x)=\\|x\\|^2$.  We set\n$\\Delta\\equiv\\Delta_\\mathcal{S}=\\oplus_X \\Delta_X$ and define $\\Delta_{\\geq X}$\nsimilarly. If $Y\\supset X$ then $\\Delta_Y=\\Delta_X\\otimes 1\n+1\\otimes\\Delta_{Y\/X}$ hence $\\Delta_{\\geq X}=\\Delta_X\\otimes 1\n+1\\otimes\\Delta_{\\mathcal{S}\/X}$.  The domain and form domain of the\noperator $\\Delta_\\mathcal{S}$ are given by $\\mathcal{H}_\\mathcal{S}^2$ and $\\mathcal{H}_\\mathcal{S}^1$ where\n$\\mathcal{H}_\\mathcal{S}^s\\equiv\\mathcal{H}^s=\\oplus_X \\mathcal{H}^s(X)$ for any real\n$s$.\n\n \nWe define nonrelativistic many-body Hamiltonian by extending to the\npresent setting \\cite[Def. 9.1]{ABG}. We consider only strictly\naffiliated operators to avoid working with not densely defined\noperators. Note that the general case of affiliated operators covers\ninteresting physical situations (hard-core interactions).\n\n\n\\begin{definition}\\label{df:NR}\n\\emph{A nonrelativistic many-body Hamiltonian of type $\\mathcal{S}$} is a\nbounded from below self-adjoint operator $H=H_\\mathcal{S}$ on $\\mathcal{H}=\\mathcal{H}_\\mathcal{S}$\nwhich is strictly affiliated to $\\mathscr{C}=\\mathscr{C}_\\mathcal{S}$ and has the following\nproperty: for each $X\\in\\mathcal{S}$ there is a bounded from below\nself-adjoint operator $H_{\\mathcal{S}\/X}$ on $\\mathcal{H}_{\\geq X}$\nsuch that \n\\begin{equation}\\label{eq:NR}\n\\mathscr{P}_{\\geq X}(H)\\equiv H_{\\geq X}=\\Delta_X\\otimes1+ 1\\otimes H_{\\mathcal{S}\/X}\n\\end{equation}\nrelatively to the tensor factorization $\\mathcal{H}_{\\geq\n  X}=\\mathcal{H}_X\\otimes\\mathcal{H}_{\\mathcal{S}\/X}$. \n\\end{definition}\n\nThen \\emph{each $H_{\\mathcal{S}\/X}$ is a nonrelativistic many-body\n  Hamiltonian of type $\\mathcal{S}\/X$}. Indeed, the argument from \\cite[p.\\\n415]{ABG} extends in a straightforward way to the present situation.\n\n\\begin{remark}\\label{re:maxs}\n  If $X$ is a maximal element in $\\mathcal{S}$ then $\\mathcal{S}\/X=\\{O\\}$ hence\n  $\\mathcal{H}_{\\mathcal{S}\/X}=\\mathcal{H}_O=\\mathbb{C}$ and $H_O$ will necessarily be a real\n  number.  Then we get $\\mathcal{H}_{\\geq X}=\\mathcal{H}_X$, $\\mathscr{C}_{\\geq X}=\\mathscr{T}_X$,\n  and $H_{\\ge X}=\\Delta_X +H_O$ on $\\mathcal{H}_X$.\n\\end{remark}\n\n\\begin{remark}\\label{re:mins}\n  Since $\\mathcal{S}$ is a finite semilattice, it has a least element\n  $\\min\\mathcal{S}$. If $\\mathcal{S}_o=\\mathcal{S}\/{\\min\\mathcal{S}}$, we get\n\\begin{equation}\\label{eq:min}\n\\mathcal{H}_{\\mathcal{S}}=\\mathcal{H}_{\\min\\mathcal{S}}\\otimes\\mathcal{H}_{\\mathcal{S}_o}, \\quad\n\\mathscr{C}_{\\mathcal{S}}=\\mathscr{T}_X\\otimes \\mathscr{C}_{\\mathcal{S}_o}, \\quad  \nH_{\\mathcal{S}}=\\Delta_{\\min\\mathcal{S}}\\otimes 1 +  1\\otimes H_{\\mathcal{S}_o}.\n\\end{equation}\n\\end{remark}\n\nNow we give an HVZ type description of the essential spectrum of a\nnonrelativistic many-body Hamiltonian. For a more detailed\nstatement, see the proof.\n\n\\begin{theorem}\\label{th:nrhvz}\n  Denote $\\tau_X=\\inf H_{\\mathcal{S}\/X}$ the bottom of the spectrum of\n  $H_{\\mathcal{S}\/X}$. Then\n\\begin{equation}\\label{eq:hvzunif}\n  \\mathrm{Sp_{ess}}(H)=[\\tau,\\infty[ \\hspace{2mm}\\text{with}\\hspace{2mm}\n  \\tau=\\min \\{ \\tau_X \\mid X \\text{ is minimal in } \\mathcal{S}\\setminus\n  \\{O\\} \\}. \n\\end{equation}\n\\end{theorem}\n\\proof\n  From \\eqref{eq:NR} we get\n\\begin{equation}\\label{eq:speint}\n\\mathrm{Sp}(H_{\\geq X})=[0,\\infty[\\ +\\, \\mathrm{Sp}(H_{\\mathcal{S}\/X})=\n[\\tau_X,\\infty[ \\quad\n\\text{if } X\\neq O.\n\\end{equation}\nIn particular, if $O\\notin\\mathcal{S}$ then by taking $X=\\min\\mathcal{S}$\nin \\eqref{eq:min} we get \n\\begin{equation}\\label{eq:mino}\n\\mathrm{Sp}(H)=\\mathrm{Sp_{ess}}(H)=[\\inf H_{\\mathcal{S}_o},\\infty[ .\n\\end{equation}\nIf $O\\in\\mathcal{S}$ then Theorem \\ref{th:imp4} implies \n\\begin{equation}\\label{eq:hvznrint}\n\\mathrm{Sp_{ess}}(H)=[\\tau,\\infty[ \\hspace{2mm}\\text{with}\\hspace{2mm}\n\\tau=\\min_{X\\in\\mathcal{P}(\\mathcal{S})} \\tau_X.\n\\end{equation}\nThe relation \\eqref{eq:hvzunif} expresses \\eqref{eq:mino} and\n\\eqref{eq:hvznrint} in a unified way. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nFor $X\\in\\mathcal{S}$ we consider the dilation group $W_\\tau=\\mathrm{e}^{i\\tau D}$\ndefined on $\\mathcal{H}_X$ by (set $n=\\dim X$):\n\\begin{equation}\\label{eq:fed}\n(W_\\tau u)(x)=\\mathrm{e}^{n\\tau\/4}u(\\mathrm{e}^{\\tau\/2}x), \n\\quad \n2iD=x\\cdot\\nabla_x+n\/2= \\nabla_x\\cdot x-n\/2. \n\\end{equation}\nLet $D_O=0$. We keep the same notation for the unitary operator\n$\\oplus_X W_\\tau$ on the direct sum $\\mathcal{H}=\\oplus_X\\mathcal{H}_X$ and we do\nnot indicate explicitly the dependence on $X$ or $\\mathcal{S}$ of $W_\\tau$\nand $D$ unless this is really needed. Note that $D$ has\nfactorization properties similar to that of the Laplacian,\ne.g. $D_{\\geq X}=D_X\\otimes 1 +1\\otimes D_{\\mathcal{S}\/X}$.\n\n\nWe refer to Subsection \\ref{ss:mest} for terminology related to the\nMourre estimate. We take $D$ as conjugate operator and we denote by\n$\\widehat\\rho_H(\\lambda)$ the best constant (which could be infinite)\nin the Mourre estimate at point $\\lambda$. The \\emph{threshold set}\n$\\tau(H)$ of $H$ with respect to $D$ is the set where\n$\\widehat\\rho_H(\\lambda)\\leq0$.  If $A$ is a real set then we define\n$N_A:\\mathbb{R}\\to[-\\infty,\\infty[$ by $N_A(\\lambda)=\\sup\\{ x\\in A \\mid\nx\\leq\\lambda\\}$ with the convention $\\sup\\emptyset=-\\infty$.  Denote\n$\\mathrm{ev}(T)$ the set of eigenvalues of an operator $T$.\n\n\\begin{theorem}\\label{th:thrintr}\n  Let $H=H_\\mathcal{S}$ be a nonrelativistic many-body Hamiltonian of type\n  $\\mathcal{S}$ and of class $C^1_\\mathrm{u}(D)$. Then $\\tau(H)$ is a closed\n  \\emph{countable} real set given by\n\\begin{equation}\\label{eq:thrintr}\n\\tau(H)=\\textstyle{\\bigcup}_{X\\neq O}\\mathrm{ev}(H_{\\mathcal{S}\/X}).\n\\end{equation}\nThe eigenvalues of $H$ which do not belong to $\\tau(H)$ are of\nfinite multiplicity and may accumulate only to points from\n$\\tau(H)$. We have\n$\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ for all real\n$\\lambda$.\n\\end{theorem}\n\nWe emphasize that if $O\\notin\\mathcal{S}$ the threshold set\n\\begin{equation}\\label{eq:thrinto}\n\\tau(H)=\\textstyle{\\bigcup}_{X\\in\\mathcal{S}} \\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\end{equation}\nis very rich although the spectrum of\n$H=\\Delta_{\\min\\mathcal{S}}\\otimes1+1\\otimes H_{\\mathcal{S}_o}$ is purely absolutely\ncontinuous.\n\n\\begin{remark}\\label{re:NM}\nWe thus see that there is no difference between nonrelativistic\n$N$-body and many-body Hamiltonians from the point of view of their\nchannel structure.  The formulas which give the essential spectrum\nand the threshold set relevant in the Mourre estimate are identical,\ncf. \\eqref{eq:hvznrint} and \\eqref{eq:thrintr}. This is due to\nthe fact that both Hamiltonian algebras are graded by the same\nsemilattice $\\mathcal{S}$.\n\\end{remark}\n\n\n\\subsection{Examples of nonrelativistic many-body Hamiltonians}\n\\label{ss:examples}\n\nLet $H=K+I$ with kinetic energy  $K=\\Delta$. Hence\n$\\mathcal{G}^1=\\mathcal{H}^1=\\oplus_X \\mathcal{H}^1_X$ and $\\mathcal{G}^{-1}=\\mathcal{H}^{-1}=\\oplus_X\n\\mathcal{H}^{-1}_X$ with the notations of \\S\\ref{ss:ex}.  The interaction\nterm is an operator $I:\\mathcal{H}^1\\to\\mathcal{H}^{-1}$ given by a sum\n$I=\\sum_{Z\\in\\mathcal{S}} I(Z)$ where each $I(Z)$ is defined with the help\nof the tensor factorization $\\mathcal{H}_{\\ge Z}=\\mathcal{H}_Z\\otimes\\mathcal{H}_{\\mathcal{S}\/Z}$.\n\n\\begin{proposition}\\label{pr:exnr}\n  Let $I^Z:\\mathcal{H}^1_{\\mathcal{S}\/Z}\\to\\mathcal{H}^{-1}_{\\mathcal{S}\/Z}$ be symmetric and small\n  at infinity and let $I(Z):=1_Z\\otimes I^Z$ which is naturally\n  defined as a symmetric operator $\\mathcal{H}^1\\to\\mathcal{H}^{-1}$. Assume that\n  $I(Z)\\geq -\\mu_Z\\Delta-\\nu$ for some numbers $\\mu_Z,\\nu\\ge0$\n  with $\\sum\\mu_Z<1$. Then $H=\\Delta+I$ defined in the quadratic\n  form sense is a nonrelativistic many-body Hamiltonian of type\n  $\\mathcal{S}$ and we have $H_{\\geq X}=\\Delta_{\\geq X}+\\sum_{Z\\supset X}\n  I(Z)$.\n\\end{proposition}\n\n\nThe first condition on $I^Z$ can be stated in terms of its\ncoefficients as follows: if $Z\\subset X\\cap Y$ then the operator\n$I_{XY}^Z:\\mathcal{H}^1_{Y\/Z}\\to\\mathcal{H}^{-1}_{X\/Z}$ is small at infinity and\nsuch that $(I_{XY}^Z)^*=I_{YX}^Z$.  On the other hand, note that if\nthe operators $I^Z:\\mathcal{H}^1_{\\mathcal{S}\/Z}\\to\\mathcal{H}^{-1}_{\\mathcal{S}\/Z}$ are compact\nthen they are small at infinity and for any $\\mu>0$ there is a\nnumber $\\nu$ such that $\\pm I(Z)\\leq \\mu\\Delta_\\mathcal{S}+\\nu$ for all\n$Z$. The more general smallness at infinity condition covers second\norder perturbations of $\\Delta_\\mathcal{S}$.\n\n\n\nIn the next proposition we give examples of nonrelativistic\noperators of class $C^1_\\mathrm{u}(D)$. The operator $H$ is constructed as\nin Proposition \\ref{pr:exnr} but we consider only interactions which\nare relatively bounded in \\emph{operator} sense with respect to the\nkinetic energy such as to force the domain of $H$ to be equal to the\ndomain of $\\Delta$, hence to $\\mathcal{H}^2=\\oplus_X\\mathcal{H}^2_X$.  Since\nthis space is stable under the action of the operators $W_\\tau$,\nwe shall get a simple condition for $H$ to be of class\n$C^1_\\mathrm{u}(D)$. \n\n\n\\begin{proposition}\\label{pr:nrm}\n  For each $Z\\in\\mathcal{S}$ assume that $I^Z:\\mathcal{H}^2_{\\mathcal{S}\/Z}\\to \\mathcal{H}_{\\mathcal{S}\/Z}$\n  is compact and symmetric as operator on $\\mathcal{H}_{\\mathcal{S}\/Z}$ and that\n  $[D,I^Z]: \\mathcal{H}^2_{\\mathcal{S}\/Z}\\to\\mathcal{H}^{-2}_{\\mathcal{S}\/Z}$ is compact.  Then the\n  conditions of Proposition \\ref{pr:exnr} are fulfilled and each\n  operator $I(Z):\\mathcal{H}^2\\to\\mathcal{H}$ is $\\Delta$-bounded with relative\n  bound zero.  The operator $H$ is self-adjoint on $\\mathcal{H}^2$ and of\n  class $C^1_\\mathrm{u}(D)$.\n\\end{proposition}\n\nSo for the coefficients $I^Z_{XY}$ we ask $I^Z_{XY}=0$ if\n$Z\\not\\subset X\\cap Y$ and if $Z\\subset X\\cap Y$ then\n$(I^{Z}_{XY})^*\\supset I^Z_{YX}$ and\n\\begin{equation}\\label{eq:3d}\nI^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{X\/Z} \\text{ and } \n[D,I^Z_{XY}]: \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\text{  are  compact\n  operators.} \n\\end{equation}\nThe expression $[D,I^Z_{XY}]=D_{X\/Z}I^Z_{XY}-I^Z_{XY}D_{Y\/Z}$ is not\nreally a commutator. Indeed, if we denote $E=(X\\cap Y)\/Z$, so\n$Y\/Z=E\\oplus(Y\/X)$ and $X\/Z=E\\oplus(X\/Y)$, then $\\mathcal{H}_{X\/Z}\n=\\mathcal{H}_E\\otimes\\mathcal{H}_{X\/Y}$ and $\\mathcal{H}_{Y\/Z} =\\mathcal{H}_E\\otimes\\mathcal{H}_{Y\/X}$.\nHence the relation $D_{X\/Z}=D_E\\otimes 1 + 1\\otimes D_{X\/Y}$ and a\nsimilar one for $Y\/Z$ give\n\\begin{equation*}\n[D,I^Z_{XY}] =[D_E,I^Z_{XY}] +D_{X\/Y}I^Z_{XY} -I^Z_{XY}D_{Y\/X}.\n\\end{equation*}\nThe first term above is a commutator and so is of a different nature\nthan the next two. Since $I^Z_{XY}D_{Y\/X}$ is a restriction of\n$(D_{Y\/X}I^Z_{YX})^*$ it is clear that the second part of condition\n\\eqref{eq:3d} follows from:\n\\begin{equation}\\label{eq:2d}\n[D_E,I^Z_{XY}] \\text{ and } D_{X\/Y}I^Z_{XY} \\text{ are compact\n  operators } \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\text{ for all } X,Y,Z.\n\\end{equation}\nWe consider some simple examples of operators $I_{XY}^Z$ to clarify\nthe difference with respect to the $N$-body situation (see\n\\S\\ref{ss:scs} for details and generalizations). If $E,F$ are\nEuclidean spaces we denote\n\\begin{equation}\\label{eq:ikef}\n\\mathscr{K}^2_{FE} =K(\\mathcal{H}^2_E,\\mathcal{H}_F) \\quad \\text{and} \\quad\n\\mathscr{K}^{2}_E=\\mathscr{K}^2_{E,E}=K(\\mathcal{H}^2_E,\\mathcal{H}_E). \n\\end{equation}\nDenote $X \\boxplus Y = X\/Y\\oplus Y\/X$ and embed $L^2(X \\boxplus\nY)\\subset \\mathscr{K}_{X\/Y,Y\/X}$ by identifying a Hilbert-Schmidt operator\nwith its kernel. Then\n\\begin{equation*}\nL^2(X \\boxplus Y;\\mathscr{K}^2_E) \\subset \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X}\n\\subset \\mathscr{K}^2_{X\/Z,Y\/Z}.\n\\end{equation*}\nThus $I_{XY}^Z \\in L^2(X \\boxplus Y;\\mathscr{K}^2_E)$ is a simple example of\noperator satisfying the first part of condition \\eqref{eq:3d}. Such\nan $I_{XY}^Z$ acts as follows: if $u\\in\\mathcal{H}^2_{Y\/Z}\\subset\nL^2(Y\/X;\\mathcal{H}^2_E)$ then\n\\[\nI_{XY}^Z u\\in \\mathcal{H}_{X\/Z} = L^2(X\/Y;\\mathcal{H}_E) \\quad\\text{is given by}\\quad\n(I_{XY}^Z u)(x')={\\textstyle\\int_{Y\/X}} I_{XY}^Z(x',y')u(y') \\text{d} y'.\n\\]\nNow we consider \\eqref{eq:2d}.\nSince $(x',y')\\mapsto[D_E,I^Z_{XY}(x',y')]$ is the kernel of the\noperator $[D_E,I^Z_{XY}]$, if\n\\[\n[D_E,I^Z_{XY}]\\in L^2(X \\boxplus Y;K(\\mathcal{H}^2_E,\\mathcal{H}^{-2}_E)\n\\]\nthen $[D_E,I^Z_{XY}]$ is a compact operator\n$\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}$. For the term $D_{X\/Y}I^Z_{XY}$ it\nsuffices to require the compactness of the operator\n\\[\nD_{X\/Y}I^Z_{XY} = 1_E\\otimes D_{X\/Y} \\cdot I^Z_{XY}: \\mathcal{H}^2_{Y\/Z}\\to\n\\mathcal{H}_E\\otimes\\mathcal{H}^{-2}_{X\/Y}.\n\\]\nFrom \\eqref{eq:fed} we see that this is a condition on the kernel\n$x'\\cdot\\nabla_{x'}I_{XY}^Z(x',y')$. For example, it suffices that\nthe operator $\\jap{Q_{X\/Y}}I^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{X\/Z}$ be\ncompact, which is a short range assumption. In summary:\n\n\\begin{example}\\label{ex:hschmidt}\n  For each $Z\\subset X\\cap Y$ let $I^Z_{XY} \\in L^2(X \\boxplus\n  Y;\\mathscr{K}^2_E)$ such that the adjoint of $I^Z_{XY}(x',y')$ is an\n  extension of $I^Z_{YX}(y',x')$. Assume that kernel\n  $[D_E,I^Z_{XY}(x',y')]$ belongs to $L^2(X \\boxplus\n  Y;K(\\mathcal{H}^2_E,\\mathcal{H}^{-2}_E)$ and that the kernel\n  $x'\\cdot\\nabla_{x'}I_{XY}^Z(x',y')$ defines a compact operator\n  $\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}$. Then \\eqref{eq:3d} is fulfilled.\n\\end{example}\n\n\\begin{example}\\label{ex:anc}\n  Here we consider the particular case $Y\\subset X$ to see the\n  structure of a generalized creation operator which appears in this\n  context. For each $Z\\subset Y$ let $I^Z_{XY}\\in\n  \\mathscr{K}^2_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}$, where the tensor product is a kind\n  of weak version of $L^2(X\/Y; \\mathscr{K}^2_{Y\/Z})$ discussed in\n  \\S\\ref{ss:ha}. Furthermore, assume that $[D_{Y\/Z},I^Z_{XY}]\\in\n  K(\\mathcal{H}^2(Y\/Z),\\mathcal{H}^{-2}_{Y\/Z})\\otimes\\mathcal{H}_{X\/Y}$ and\n  $D_{X\/Y}I^Z_{XY}\\in \\mathscr{K}^2_{Y\/Z}\\otimes\\mathcal{H}^{-2}_{X\/Y}$. Then\n  \\eqref{eq:3d} holds.\n\\end{example}\n\n\n\\subsection{Boundary values of the resolvent}\n\\label{ss:bvr}\n\nTheorem \\ref{th:thrintr} has important consequences in the spectral\ntheory of the operator $H$: we shall use it together with\n\\cite[Theorem 7.4.1]{ABG} to show that $H$ has no singular\ncontinuous spectrum and to prove the existence of the boundary\nvalues of its resolvent in the class of weighted $L^2$ spaces that\nwe define now. Let $\\mathcal{H}_{s,p}=\\oplus_X L^2_{s,p}(X)$ where the\n$L^2_{s,p}(X)$ are the Besov spaces associated to the position\nobservable on $X$ (these are obtained from the usual Besov spaces\nassociated to $L^2(X)$ by a Fourier transformation).  Note that\n$\\mathcal{H}_{s}=\\mathcal{H}_{s,2}$ is the Fourier transform of the Sobolev space\n$\\mathcal{H}^s$.  Let $\\mathbb{C}_+$ be the open upper half plane and\n$\\mathbb{C}^H_+=\\mathbb{C}_+\\cup(\\mathbb{R}\\setminus\\tau(H))$. If we replace the upper\nhalf plane by the lower one we similarly get the sets $\\mathbb{C}_-$ and\n$\\mathbb{C}^H_-$. We define two holomorphic maps $R_\\pm:\\mathbb{C}_\\pm\\to\nL(\\mathcal{H})$ by $R_\\pm(z)=(H-z)^{-1}$ and note that we have continuous\nembeddings\n\\[\nL(\\mathcal{H})\\subset L(\\mathcal{H}_{1\/2,1},\\mathcal{H}_{-1\/2,\\infty}) \\subset\nL(\\mathcal{H}_{s},\\mathcal{H}_{-s}) \\quad\\text{if } s>1\/2\n\\]\nso we may consider $R_\\pm$ as maps with values in\n$L(\\mathcal{H}_{1\/2,1},\\mathcal{H}_{-1\/2,\\infty})$.\n\n\\begin{theorem}\\label{th:c11}\nIf $H$ is of class $C^{1,1}(D)$ then its singular continuous\nspectrum is empty and the holomorphic maps $R_\\pm:\\mathbb{C}_\\pm\\to\nL(\\mathcal{H}_{1\/2,1},\\mathcal{H}_{-1\/2,\\infty})$ extend to weak$^*$ continuous\nfunctions $\\bar{R}_\\pm$ on $\\mathbb{C}^H_\\pm$.  The maps\n$\\bar{R}_\\pm:\\mathbb{C}^H_\\pm\\to L(\\mathcal{H}_{s},\\mathcal{H}_{-s})$ are norm continuous\nif $s>1\/2$.\n\\end{theorem}\n\nThis result is optimal both with regard to the regularity of the\nHamiltonian relatively to the conjugate operator $D$ and to the\nBesov spaces in which we establish the existence of the boundary\nvalues of the resolvent. The class $C^{1,1}(D)$ will be discussed\nand its optimality will be made precise in \\S\\ref{ss:scsc} but we\ngive some examples below.\n\nWe state first the simplest sufficient condition: \\emph{assume that\n  $H$ is as in Proposition \\ref{pr:exnr} and that its domain is\n  equal to $\\mathcal{H}^2$;if $\\,[D,[D,I^Z]]\\in\n  L(\\mathcal{H}^2_{\\mathcal{S}\/Z},\\mathcal{H}^{-2}_{\\mathcal{S}\/Z})$ for all $Z$ then $H$ is of\n  class $C^{1,1}(D)$}. This follows from Theorem 6.3.4 in\n\\cite{ABG}. The condition on $\\,[D,[D,I^Z]]$ can easily be written\nin terms of the coefficients $I^Z_{XY}$ by arguments similar to\nthose of \\S\\ref{ss:examples}.  Refinements allow the addition of\nlong range and short range interactions as in \\cite[\\S 9.4.2]{ABG}.\n\nLet $\\xi:\\mathbb{R}\\to\\mathbb{R}$ be of class $C^\\infty$ and such that\n$\\xi(\\lambda)=0$ if $\\lambda\\le 1$ and $\\xi(\\lambda)=1$ if\n$\\lambda\\ge 2$. For each Euclidean space $X$ and real $r\\ge 1$ we\ndenote $\\xi^r_X$ the operator of multiplication by the function\n$x\\mapsto\\xi(|x|\/r)$ on any Sobolev space over $X$. Then we define\n$\\xi^r_\\mathcal{S}=\\oplus_{X\\in\\mathcal{S}}\\xi^r_X$ considered as operator on\n$\\mathcal{H}^s_\\mathcal{S}$ for any real $s$.\n\n\\begin{definition}\\label{df:slr}\n  Let $T:\\mathcal{H}^2_\\mathcal{S}\\to\\mathcal{H}_\\mathcal{S}$ be a symmetric operator. We say that\n  $T$ is a \\emph{long range interaction} if $[D,T]\\in\n  L(\\mathcal{H}^2_\\mathcal{S},\\mathcal{H}^{-1}_\\mathcal{S})$ and $\\int_1^\\infty \\|\\xi^r_\\mathcal{S}\n  [D,T]\\|_{\\mathcal{H}^2_\\mathcal{S}\\to\\mathcal{H}^{-1}_\\mathcal{S}} \\text{d} r\/r <\\infty$.  We say that\n  $T$ is a \\emph{short range interaction} if $\\int_1^\\infty\n  \\|\\xi^r_\\mathcal{S} [D,T]\\|_{\\mathcal{H}^2_\\mathcal{S}\\to\\mathcal{H}_\\mathcal{S}} \\text{d} r <\\infty$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{th:BVR}\n  Assume that $H=\\Delta_\\mathcal{S} + \\sum_{Z\\in\\mathcal{S}} 1_Z\\otimes I^Z$ where\n  each $I^z:\\mathcal{H}^2_{\\mathcal{S}\/Z}\\to\\mathcal{H}_{\\mathcal{S}\/Z}$ is symmetric, compact, and\n  is the sum of a long range and a short range interaction. Then $H$\n  is a nonrelativistic many-body Hamiltonian of class $C^{1,1}(D)$,\n  hence the conclusions of Theorem \\ref{th:c11} are true.\n\\end{theorem}\n\nScattering channels may be defined in a natural way in the context\nof the theorem.  If the long range interactions are absent we expect\nthat asymptotic completeness holds.\n\n\n\n\n\n\\section{Graded Hilbert $C^*$-modules}\n\\label{s:grad}\n\\protect\\setcounter{equation}{0}\n\n\\subsection{Graded $\\boldsymbol{C^*}$-algebras}\n\\label{ss:grca}\n\nThe natural framework for the systems considered in this paper is\nthat of $C^*$-algebras graded by semilattices. We refer to\n\\cite{Ma2,Ma3} for a detailed study of this class of algebras.\n\nLet $\\mathcal{S}$ be a semilattice and $\\mathscr{A}$ a graded $C^*$-algebra.\nFollowing \\cite{Ma2} we say that $\\mathscr{B}\\subset\\mathscr{A}$ is a \\emph{graded\n  $C^*$-subalgebra} if $\\mathscr{B}$ is a $C^*$-subalgebra of $\\mathscr{A}$ equal to\n$\\sum^\\mathrm{c}_\\sigma\\mathscr{B}\\cap\\mathscr{A}(\\sigma)$. Then $\\mathscr{B}$ has a natural\ngraded $C^*$-algebra structure: $\\mathscr{B}(\\sigma)=\\mathscr{B}\\cap\\mathscr{A}(\\sigma)$.\nIf $\\mathscr{B}$ is also an ideal of $\\mathscr{A}$ then $\\mathscr{B}$ is a \\emph{graded\n  ideal}.\n\nA subset $\\mathcal{T}$ of a semilattice $\\mathcal{S}$ is a \\emph{sub-semilattice of\n  $\\mathcal{S}$} if $\\sigma,\\tau\\in\\mathcal{T} \\Rightarrow\n\\sigma\\wedge\\tau\\in\\mathcal{T}$. We say that $\\mathcal{T}$ is an \\emph{ideal of\n  $\\mathcal{S}$} if $\\sigma\\leq\\tau\\in\\mathcal{T} \\Rightarrow \\sigma\\in\\mathcal{T}$. If\n$\\mathscr{A}$ is an $\\mathcal{S}$-graded $C^*$-algebra and $\\mathcal{T}\\subset\\mathcal{S}$ let\n$\\mathscr{A}(\\mathcal{T})=\\sum^\\mathrm{c}_{\\sigma\\in\\mathcal{T}}\\mathscr{A}(\\sigma)$ (if $\\mathcal{T}$ is finite\nthe sum is already closed).  If $\\mathcal{T}$ is a sub-semilattice or an\nideal then clearly $\\mathscr{A}(\\mathcal{T})$ is a $C^*$-subalgebra or an ideal of\n$\\mathscr{A}$ respectively.\n\nWe say that $\\mathscr{A}$ is \\emph{supported by a sub-semilattice $\\mathcal{T}$} if\n$\\mathscr{A}=\\mathscr{A}(\\mathcal{T})$, i.e. $\\mathscr{A}(\\sigma)=\\{0\\}$ for $\\sigma\\notin\\mathcal{T}$. Then\n$\\mathscr{A}$ is also $\\mathcal{T}$-graded.  The smallest sub-semilattice with this\nproperty will be called \\emph{support of $\\mathscr{A}$}.  If $\\mathcal{T}$ is a\nsub-semilattice of $\\mathcal{S}$ and $\\mathscr{A}$ is a $\\mathcal{T}$-graded algebra then\n$\\mathscr{A}$ is $\\mathcal{S}$-graded: set $\\mathscr{A}(\\sigma)=\\{0\\}$ for\n$\\sigma\\in\\mathcal{S}\\setminus\\mathcal{T}$.\n\nThe next result is obvious if $\\mathcal{S}$ is finite. For the general case,\nsee the proof of Proposition 3.3 in \\cite{DG3}.\n\n\n\\begin{proposition}\\label{pr:gsalg}\n  Let $\\mathcal{T}$ be a sub-semilattice of $\\mathcal{S}$ such that\n  $\\mathcal{T}'=\\mathcal{S}\\setminus\\mathcal{T}$ is an ideal. Then $\\mathscr{A}(\\mathcal{T})$ is a\n  $C^*$-subalgebra of $\\mathscr{A}$, $\\mathscr{A}(\\mathcal{T}')$ is an ideal of $\\mathscr{A}$, and\n  $\\mathscr{A}=\\mathscr{A}(\\mathcal{T})+\\mathscr{A}(\\mathcal{T}')$ with $\\mathscr{A}(\\mathcal{T})\\cap\\mathscr{A}(\\mathcal{T}')=\\{0\\}$.\n  In particular, the natural linear projection\n  $\\mathscr{P}(\\mathcal{T}):\\mathscr{A}\\to\\mathscr{A}(\\mathcal{T})$ is a morphism.\n\\end{proposition}\n\nIf $\\mathcal{T}$ is a sub-semilattice then $\\mathcal{T}'$ is an ideal if and only if\n$\\mathcal{T}$ is a filter\n(i.e. $\\sigma\\ge\\tau\\in\\mathcal{T}\\Rightarrow\\sigma\\in\\mathcal{T}$). Thus if $\\mathcal{S}$\nis finite then the only sub-semilattices which have this property\nare the $\\mathcal{S}_{\\ge\\sigma}$ introduced below.\n\nThe simplest sub-semilattices are the chains (totally ordered\nsubsets). If $\\sigma\\in \\mathcal{S}$ and\n\\begin{equation}\\label{eq:Aa}\n\\mathcal{S}_{\\geq\\sigma}=\\{\\tau\\in \\mathcal{S}\\mid \\tau\\geq\\sigma\\},\n\\quad%\n\\mathcal{S}_{\\not\\geq\\sigma}=\n\\mathcal{S}'_{\\geq\\sigma}=\\{\\tau\\in\\mathcal{S}\\mid\\tau\\not\\geq\\sigma\\},\n\\quad\n\\mathcal{S}_{\\leq\\sigma}=\\{\\tau\\in \\mathcal{S}\\mid \\tau\\leq\\sigma\\}\n\\end{equation}\nthen $\\mathcal{S}_{\\geq\\sigma}$ is a sub-semilattice and\n$\\mathcal{S}_{\\not\\geq\\sigma}$ and $\\mathcal{S}_{\\leq\\sigma}$ are ideals. So\n$\\mathscr{A}_{\\ge\\sigma}\\equiv\\mathscr{A}(\\mathcal{S}_{\\geq\\sigma})$ is a graded\n$C^*$-subalgebra of $\\mathscr{A}$ supported by $\\mathcal{S}_{\\geq\\sigma}$ and\n$\\mathscr{A}(\\mathcal{S}_{\\not\\geq\\sigma})$ is a graded ideal supported by\n$\\mathcal{S}_{\\not\\geq\\sigma}$ such that\n\\begin{equation}\\label{eq:dsum}\n\\mathscr{A}=\\mathscr{A}_{\\ge\\sigma}+\\mathscr{A}(\\mathcal{S}_{\\not\\geq\\sigma})\n\\quad\\text{with} \\quad\n\\mathscr{A}_{\\ge\\sigma}\\cap\\mathscr{A}(\\mathcal{S}_{\\not\\geq\\sigma})=\\{0\\}. \n\\end{equation}\nThe projection morphism $\\mathscr{P}_{\\ge\\sigma}:\\mathscr{A}\\to\\mathscr{A}_{\\geq\\sigma}$\ndefined by \\eqref{eq:dsum} is the unique linear continuous map\n$\\mathscr{P}_{\\geq\\sigma}:\\mathscr{A}\\rightarrow\\mathscr{A}$ such that $\\mathscr{P}_{\\geq\\sigma}A=A$ if\n$A\\in\\mathscr{A}(\\tau)$ for some $\\tau\\geq\\sigma$ and $\\mathscr{P}_{\\geq\\sigma}A=0$\notherwise.\n\n$\\mathcal{S}$ is called \\emph{atomic} if it has a smallest element\n$o\\equiv\\min \\mathcal{S}$ and if each $\\sigma\\neq o$ is minorated by an\natom.  We denote by $\\mathcal{P}(\\mathcal{S})$ the set of atoms of $\\mathcal{S}$.  If $\\mathcal{T}$\nis an ideal of $\\mathcal{S}$ and $\\mathcal{S}$ is atomic then $\\mathcal{T}$ is atomic, we\nhave $\\min\\mathcal{T}=\\min \\mathcal{S}$, and $\\mathcal{P}(\\mathcal{T})=\\mathcal{P}(\\mathcal{S})\\cap\\mathcal{T}$.  This next\nresult is also easy to prove \\cite{DG3}.\n\n\\begin{theorem}\\label{th:ga}  \n  If $\\mathcal{S}$ is atomic then $\\mathscr{P}\n  A=(\\mathscr{P}_{\\geq\\alpha}A)_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}$ defines a morphism\n  $\\mathscr{P}:\\mathscr{A}\\to\\prod_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}\\mathscr{A}_{\\geq\\alpha}$ with\n  $\\mathscr{A}(o)$ as kernel. This gives us a canonical embedding\n\\begin{equation}\\label{eq:quot}\n\\mathscr{A}\/\\mathscr{A}(o)\\subset\\displaystyle\\mbox{$\\textstyle\\prod$}_{\\substack{i}}_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}\\mathscr{A}_{\\geq\\alpha}.\n\\end{equation}\n\\end{theorem}\n\nWe call this ``theorem'' because it has important consequences in\nthe spectral theory of many-body Hamiltonians: it allows us to\ncompute their essential spectrum and to prove the Mourre estimate.\n\nWe assume that $\\mathcal{S}$ is atomic so that $\\mathscr{A}$ comes equipped with a\nremarkable ideal $\\mathscr{A}(o)$. Then for $A\\in\\mathscr{A}$ we define its\n\\emph{essential spectrum} (relatively to $\\mathscr{A}(o)$) by the formula\n\\begin{equation}\\label{eq:eso}\n\\mathrm{Sp_{ess}}(A)\\equiv\\mathrm{Sp}(\\mathscr{P} A).\n\\end{equation}\nIn our concrete examples $\\mathscr{A}$ is represented on a Hilbert space\n$\\mathcal{H}$ and $\\mathscr{A}(o)= K(\\mathcal{H})$, so we get the usual Hilbertian notion of\nessential spectrum. \n\nIn order to extend this to unbounded operators it is convenient to\ndefine an \\emph{observable affiliated to $\\mathscr{A}$} as a morphism\n$H:\\cc_{\\mathrm{o}}(\\mathbb{R})\\to\\mathscr{A}$.  We set $\\varphi(H)\\equiv H(\\varphi)$.  If\n$\\mathscr{A}$ is realized on  $\\mathcal{H}$ then a self-adjoint\noperator on $\\mathcal{H}$ such that $(H+i)^{-1}\\in\\mathscr{A}$ is said to be\naffiliated to $\\mathscr{A}$; then $H(\\varphi)=\\varphi(H)$ defines an\nobservable affiliated to $\\mathscr{A}$ (see Appendix A in \\cite{DG3} for a\nprecise description of the relation between observables and\nself-adjoint operators affiliated to $\\mathscr{A}$). The spectrum of an\nobservable is by definition the support of the morphism $H$:\n\\begin{equation}\\label{eq:sp}\n\\mathrm{Sp}(H)=\\{\\lambda\\in\\mathbb{R} \\mid\n\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R}),\\varphi(\\lambda)\\neq 0 \\Rightarrow\n\\varphi(H)\\neq0\\}. \n\\end{equation}\nNow note that $\\mathscr{P} H\\equiv\\mathscr{P}\\circ H$ is an observable affiliated to\nthe quotient algebra $\\mathscr{A}\/\\mathscr{A}(o)$ so we may define the essential\nspectrum of $H$ as the spectrum of $\\mathscr{P} H$. Explicitly, we get:\n\\begin{equation}\\label{eq:es1}\n\\mathrm{Sp_{ess}}(H)=\\{\\lambda\\in\\mathbb{R} \\mid \n\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R}),\\varphi(\\lambda)\\neq 0 \\Rightarrow\n\\varphi(H)\\notin \\mathscr{A}(o)\\}. \n\\end{equation}\nNow the first assertion of the next theorem follows immediately from\nTheorem \\ref{th:ga}. For the second assertion, see the proof of\nTheorem 2.10 in \\cite{DG2}. By $\\overline{\\cup}$ we denote the\nclosure of the union.\n\n\\begin{theorem}\\label{th:gas}\nLet $\\mathcal{S}$ be atomic. If $H$ is an observable affiliated to $\\mathscr{A}$\nthen $H_{\\geq\\alpha}=\\mathscr{P}_{\\geq\\alpha}H$ is an observable affiliated\nto $\\mathscr{A}_{\\geq\\alpha}$ and we have:\n\\begin{equation}\\label{eq:es2}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{\\alpha\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq\\alpha}).\n\\end{equation}\nIf for each $A\\in\\mathscr{A}$ the set of $\\mathscr{P}_{\\geq\\alpha}A$ with\n$\\alpha\\in\\mathcal{P}(\\mathcal{S})$ is compact in $\\mathscr{A}$ then the union in\n\\eqref{eq:es2} is closed.\n\\end{theorem}\n\n\\subsection{Hilbert $C^*$-modules}\n\\label{ss:preh}\n\nSome basic knowledge of the theory of Hilbert\n$C^*$-modules is useful but not indispensable for understanding our\nconstructions. We translate here the necessary facts in a purely\nHilbert space language.  Our main reference for the general theory\nof Hilbert $C^*$-modules is \\cite{La} but see also \\cite{Bl,RW}.\nThe examples of interest in this paper are the ``concrete'' Hilbert\n$C^*$-modules described below as Hilbert $C^*$-submodules of\n$L(\\mathcal{E},\\mathcal{F})$. We recall, however, the general definition.\n\n\nIf $\\mathscr{A}$ is a $C^*$-algebra then a \\emph{Banach $\\mathscr{A}$-module} is a\nBanach space $\\mathscr{M}$ equipped with a continuous bilinear map\n$\\mathscr{A}\\times\\mathscr{M}\\ni(A,M)\\mapsto MA\\in\\mathscr{M}$ such that $(MA)B=M(AB)$.  We\ndenote $\\mathscr{M}\\cdot\\mathscr{A}$ the clspan of the elements $MA$ with $A\\in\\mathscr{A}$\nand $M\\in\\mathscr{M}$.  By the Cohen-Hewitt theorem \\cite{FD} for each\n$N\\in\\mathscr{M}\\cdot\\mathscr{A}$ there are $A\\in\\mathscr{A}$ and $M\\in\\mathscr{M}$ such that\n$N=MA$, in particular $\\mathscr{M}\\cdot\\mathscr{A}=\\mathscr{M}\\mathscr{A}$.  Note that by module we\nmean ``right module'' but the Cohen-Hewitt theorem is also valid for\nleft Banach modules.\n\nLet $\\mathscr{A}$ be a $C^*$-algebra. A (right) \\emph{Hilbert $\\mathscr{A}$-module}\nis a Banach $\\mathscr{A}$-module $\\mathscr{M}$ equipped with an $\\mathscr{A}$-valued\nsesquilinear map\n$\\braket{\\cdot}{\\cdot}\\equiv\\braket{\\cdot}{\\cdot}_\\mathscr{A}$ which is\npositive (i.e. $\\braket{M}{M}\\geq0$) $\\mathscr{A}$-sesquilinear\n(i.e. $\\braket{M}{NA}=\\braket{M}{N}A$) and such that\n$\\|M\\|\\equiv\\|\\braket{M}{M}\\|^{1\/2}$. Then $\\mathscr{M}=\\mathscr{M}\\mathscr{A}$.  The clspan\nof the elements $\\braket{M}{M}$ is an ideal of $\\mathscr{A}$ denoted\n$\\braket{\\mathscr{M}}{\\mathscr{M}}$. One says that $\\mathscr{M}$ is \\emph{full} if\n$\\braket{\\mathscr{M}}{\\mathscr{M}}=\\mathscr{A}$.  If $\\mathscr{A}$ is an ideal of a $C^*$-algebra\n$\\mathscr{C}$ then $\\mathscr{M}$ is equipped with an obvious structure of Hilbert\n$\\mathscr{C}$-module. Left Hilbert $\\mathscr{A}$-modules are defined similarly. \n\nIf $\\mathscr{M},\\mathscr{N}$ are Hilbert $\\mathscr{A}$-modules and $(M,N)\\in\\mathscr{M}\\times\\mathscr{N}$\nthen $M'\\mapsto N\\braket{M}{M'}$ is a linear continuous map\n$\\mathscr{M}\\to\\mathscr{N}$ denoted $\\ket{N}\\bra{M}$ or $NM^*$.  The closed linear\nsubspace of $L(\\mathscr{M},\\mathscr{N})$ generated by these elements is denoted\n$\\mathcal{K}(\\mathscr{M},\\mathscr{N})$. There is a unique antilinear isometric map $T\\mapsto\nT^*$ of $\\mathcal{K}(\\mathscr{M},\\mathscr{N})$ onto $\\mathcal{K}(\\mathscr{N},\\mathscr{M})$ which sends\n$\\ket{N}\\bra{M}$ into $\\ket{M}\\bra{N}$. The space\n$\\mathcal{K}(\\mathscr{M})\\equiv\\mathcal{K}(\\mathscr{M},\\mathscr{M})$ is a $C^*$-algebra called\n\\emph{imprimitivity algebra} of the Hilbert $\\mathscr{A}$-module $\\mathscr{M}$.\n\nAssume that $\\mathscr{N}$ is a closed subspace of a Hilbert $\\mathscr{A}$-module $\\mathscr{M}$\nand let $\\braket{\\mathscr{N}}{\\mathscr{N}}$ be the clspan of the elements\n$\\braket{N}{N}$ in $\\mathscr{A}$. If $\\mathscr{N}$ is an $\\mathscr{A}$-submodule of $\\mathscr{M}$ then\nit inherits an obvious Hilbert $\\mathscr{A}$-module structure from $\\mathscr{M}$.  If\n$\\mathscr{N}$ is not an $\\mathscr{A}$-submodule of $\\mathscr{M}$ it may happen that there is a\n$C^*$-subalgebra $\\mathscr{B}\\subset\\mathscr{A}$ such that $\\mathscr{N}\\mathscr{B}\\subset\\mathscr{N}$ and\n$\\braket{\\mathscr{N}}{\\mathscr{N}}\\subset\\mathscr{B}$. Then clearly we get a Hilbert\n$\\mathscr{B}$-module structure on $\\mathscr{N}$. On the other hand, it is clear that\nsuch a $\\mathscr{B}$ exists if and only if $\\mathscr{N}\\braket{\\mathscr{N}}{\\mathscr{N}}\\subset\\mathscr{N}$\nand then $\\braket{\\mathscr{N}}{\\mathscr{N}}$ is a $C^*$-subalgebra of $\\mathscr{A}$. Under\nthese conditions we say that \\emph{$\\mathscr{N}$ is a Hilbert $C^*$-submodule}\nof the Hilbert $\\mathscr{A}$-module $\\mathscr{M}$. Then $\\mathscr{N}$ inherits a Hilbert\n$\\braket{\\mathscr{N}}{\\mathscr{N}}$-module structure and this defines the\n$C^*$-algebra $\\mathcal{K}(\\mathscr{N})$. Moreover, if $\\mathscr{B}$ is as above then\n$\\mathcal{K}(\\mathscr{N})=\\mathcal{K}_\\mathscr{B}(\\mathscr{N})$.\n\nIf $\\mathscr{N}$ is a closed subspace of a Hilbert $\\mathscr{A}$-module $\\mathscr{M}$ then\nlet $\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$ be the closed subspace of $\\mathcal{K}(\\mathscr{M})$ generated by\nthe elements $NN^*$ with $N\\in\\mathscr{N}$. It is easy to prove that\n\\emph{if $\\mathscr{N}$ is a Hilbert $C^*$-submodule of $\\mathscr{M}$ then\n$\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$ is a $C^*$-subalgebra of $\\mathcal{K}(\\mathscr{M})$ and the map\n$T\\mapsto T|_\\mathscr{N}$ sends $\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$ onto $\\mathcal{K}(\\mathscr{N})$ and is an\nisomorphism of $C^*$-algebras}. Then we identify $\\mathcal{K}(\\mathscr{N}|\\mathscr{M})$\nwith $\\mathcal{K}(\\mathscr{N})$.\n\nIf $\\mathcal{E},\\mathcal{F}$ are Hilbert spaces then we equip $L(\\mathcal{E},\\mathcal{F})$ with the\nHilbert $L(\\mathcal{E})$-module structure defined as follows: the\n$C^*$-algebra $L(\\mathcal{E})$ acts to the right by composition and we take\n$\\braket{M}{N}=M^*N$ as inner product, where $M^*$ is the usual\nadjoint of the operator $M$. Note that $L(\\mathcal{E},\\mathcal{F})$ is also equipped\nwith a natural left Hilbert $L(\\mathcal{F})$-module structure: this time the\ninner product is $MN^*$.\n\nIf $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$ is a linear subspace then $\\mathscr{M}^*\\subset\nL(\\mathcal{F},\\mathcal{E})$ is the set of adjoint operators $M^*$ with $M\\in\\mathscr{M}$.\nClearly $\\mathscr{M}_1\\subset\\mathscr{M}_2\\Rightarrow\\mathscr{M}_1^*\\subset\\mathscr{M}_2^*$. If\n$\\mathcal{G}$ is a third Hilbert spaces and $\\mathscr{N}\\subset L(\\mathcal{F},\\mathcal{G})$ is a\nlinear subspace then $(\\mathscr{N}\\cdot\\mathscr{M})^*=\\mathscr{M}^*\\cdot\\mathscr{N}^*$.  In\nparticular, if $\\mathcal{E}=\\mathcal{F}=\\mathcal{G}$, $\\mathscr{M}=\\mathscr{M}^*$, and $\\mathscr{N}=\\mathscr{N}^*$ then\n$\\mathscr{M}\\cdot\\mathscr{N}\\subset\\mathscr{N}\\cdot\\mathscr{M}$ is equivalent to\n$\\mathscr{M}\\cdot\\mathscr{N}=\\mathscr{N}\\cdot\\mathscr{M}$.\n\n\nNow let $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$ be a closed linear subspace. Then\n\\emph{$\\mathscr{M}$ is a Hilbert $C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$ if and only\n  if $\\mathscr{M}\\mr^*\\mathscr{M}\\subset\\mathscr{M}$}.\n\nThese are the ``concrete'' Hilbert $C^*$-modules we are interested\nin. It is clear that $\\mathscr{M}^*$ will be a Hilbert $C^*$-submodule of\n$L(\\mathcal{F},\\mathcal{E})$. We mention that $\\mathscr{M}^*$ is canonically identified with\nthe left Hilbert $\\mathscr{A}$-module $\\mathcal{K}(\\mathscr{M},\\mathscr{A})$ dual to $\\mathscr{M}$. \n\n\\begin{proposition}\\label{pr:ss}\nLet $\\mathcal{E},\\mathcal{F}$ be Hilbert spaces and let $\\mathscr{M}$ be a Hilbert\n$C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$. Then $\\mathscr{A}\\equiv\\mathscr{M}^*\\cdot\\mathscr{M}$ and\n$\\mathscr{B}\\equiv\\mathscr{M}\\cdot\\mathscr{M}^*$ are $C^*$-algebras of operators on $\\mathcal{E}$\nand $\\mathcal{F}$ respectively and $\\mathscr{M}$ is equipped with a canonical\nstructure of $(\\mathscr{B},\\mathscr{A})$ imprimitivity bimodule.\n\\end{proposition}\n\nFor the needs of this paper the last assertion of the proposition\ncould be interpreted as a definition.\n\n\\begin{proposition}\\label{pr:clsubmod}\nLet $\\mathscr{N}$ be a $C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$ such that\n$\\mathscr{N}\\subset\\mathscr{M}$ and $\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{M}^*\\cdot\\mathscr{M}$,\n$\\mathscr{N}\\cdot\\mathscr{N}^*=\\mathscr{M}\\cdot\\mathscr{M}^*$. Then $\\mathscr{N}=\\mathscr{M}$.\n\\end{proposition}\n\\proof If $M\\in\\mathscr{M}$ and $N\\in\\mathscr{N}$ then $MN^*\\in\\mathscr{B}=\\mathscr{N}\\cdot\\mathscr{N}^*$ and\n$\\mathscr{N}\\rn^*\\mathscr{N}\\subset\\mathscr{N}$ hence $MN^*N\\in\\mathscr{N}$. Since $\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{A}$\nwe get $MA\\in\\mathscr{N}$ for all $A\\in\\mathscr{A}$. Let $A_i$ be an approximate\nidentity for the $C^*$-algebra $\\mathscr{A}$. Since one can factorize $M=M'A'$\nwith $M'\\in\\mathscr{M}$ and $A'\\in\\mathscr{A}$ the sequence $MA_i=M'A'A_i$ converges\nto $ M'A'=M$ in norm. Thus $M\\in\\mathscr{N}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{proposition}\\label{pr:2ss}\nLet $\\mathcal{E},\\mathcal{F},\\mathcal{H}$ be Hilbert spaces and let $\\mathscr{M}\\subset L(\\mathcal{H},\\mathcal{E})$\nand $\\mathscr{N}\\subset L(\\mathcal{H},\\mathcal{F})$ be Hilbert $C^*$-submodules. Let $\\mathscr{A}$ be\na $C^*$-algebra of operators on $\\mathcal{H}$ such that $\\mathscr{M}^*\\cdot\\mathscr{M}$ and\n$\\mathscr{N}^*\\cdot\\mathscr{N}$ are ideals of $\\mathscr{A}$ and let us view $\\mathscr{M}$ and $\\mathscr{N}$ as\nHilbert $\\mathscr{A}$-modules. Then $\\mathcal{K}(\\mathscr{M},\\mathscr{N})\\cong\\mathscr{N}\\cdot\\mathscr{M}^*$ the\nisometric isomorphism being determined by the condition\n$\\ket{N}\\bra{M}=NM^*$. \n\\end{proposition}\n\n\\subsection{Graded Hilbert $C^*$-modules}\n\\label{ss:gf}\n\nThis is due to Georges Skandalis \\cite{Sk} (see also Remark\n\\ref{re:squant}).\n\n\\begin{definition}\\label{df:grm}\nLet $\\mathcal{S}$ be a semilattice and $\\mathscr{A}$ an $\\mathcal{S}$-graded\n$C^*$-algebra. A Hilbert $\\mathscr{A}$-module $\\mathscr{M}$ is an \\emph{$\\mathcal{S}$-graded\n  Hilbert $\\mathscr{A}$-module} if a linearly independent family\n$\\{\\mathscr{M}(\\sigma)\\}_{\\sigma\\in \\mathcal{S}}$ of closed subspaces of $\\mathscr{M}$ is\ngiven such that $\\sum_\\sigma\\mathscr{M}(\\sigma)$ is dense in $\\mathscr{M}$ and:\n\\begin{equation}\\label{eq:grm}\n\\mathscr{M}(\\sigma)\\mathscr{A}(\\tau)\\subset\\mathscr{M}(\\sigma\\wedge\\tau) \n\\hspace{2mm}\\text{and}\\hspace{2mm}\n\\braket{\\mathscr{M}(\\sigma)}{\\mathscr{M}(\\tau)}\\subset\\mathscr{A}(\\sigma\\wedge\\tau)\n\\hspace{2mm} \\text{for all } \\sigma,\\tau\\in \\mathcal{S}.\n\\end{equation}\n\\end{definition}\nNote that $\\mathscr{A}$ equipped with its canonical Hilbert $\\mathscr{A}$-module\nstructure is an $\\mathcal{S}$-graded Hilbert \\mbox{$\\mathscr{A}$-module}. \n\\eqref{eq:grm} implies that each $\\mathscr{M}(\\sigma)$ is a\nHilbert $\\mathscr{A}(\\sigma)$-module and if $\\sigma\\leq\\tau$ then\n$\\mathscr{M}(\\sigma)$ is an $\\mathscr{A}(\\tau)$-module.\n\n\nFrom \\eqref{eq:grm} we also see that \\emph{the imprimitivity algebra\n  $\\mathcal{K}(\\mathscr{M}(\\sigma))$ of the Hilbert $\\mathscr{A}(\\sigma)$-module\n  $\\mathscr{M}(\\sigma)$ is naturally identified with the clspan in\n  $\\mathcal{K}(\\mathscr{M})$ of the elements $MM^*$ with $M\\in\\mathscr{M}(\\sigma)$}. Thus\n$\\mathcal{K}(\\mathscr{M}(\\sigma))$ is identified with a $C^*$-subalgebra of\n$\\mathcal{K}(\\mathscr{M})$. We use this identification below.\n\n\\begin{theorem}\\label{th:kghm}\nIf $\\mathscr{M}$ is a graded Hilbert $\\mathscr{A}$-module then $\\mathcal{K}(\\mathscr{M})$ becomes a\ngraded $C^*$-algebra if we define\n$\\mathcal{K}(\\mathscr{M})(\\sigma)=\\mathcal{K}(\\mathscr{M}(\\sigma))$. If $M\\in\\mathscr{M}(\\sigma)$ and\n$N\\in\\mathscr{M}(\\tau)$ then there are elements $M'$ and $N'$ in\n$\\mathscr{M}(\\sigma\\wedge\\tau)$ such that $MN^*=M'N'^*$;\nin particular $MN^*\\in\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$.\n\\end{theorem}\n\\proof As explained before, $\\mathcal{K}(\\mathscr{M})(\\sigma)$ are $C^*$-subalgebras\nof $\\mathcal{K}(\\mathscr{M})$. To show that they are linearly independent, let\n$T(\\sigma)\\in\\mathcal{K}(\\mathscr{M})(\\sigma)$ such that $T(\\sigma)=0$ but for a\nfinite number of $\\sigma$ and assume $\\sum_\\sigma T(\\sigma)=0$. Then\nfor each $M\\in\\mathscr{M}$ we have $\\sum_\\sigma T(\\sigma)M=0$. Note that the\nrange of $T(\\sigma)$ is included in $\\mathscr{M}(\\sigma)$. Since the linear\nspaces $\\mathscr{M}(\\sigma)$ are linearly independent we get $T(\\sigma)M=0$\nfor all $\\sigma$ and $M$ hence $T(\\sigma)=0$ for all $\\sigma$.\n\nWe now prove the second assertion of the proposition. Since\n$\\mathscr{M}(\\sigma)$ is a Hilbert $\\mathscr{A}(\\sigma)$-module there are\n$M_1\\in\\mathscr{M}(\\sigma)$ and $S\\in\\mathscr{A}(\\sigma)$ such that $M=M_1S$, cf. the\nCohen-Hewitt theorem or Lemma 4.4 in \\cite{La}. Similarly, $N=N_1T$\nwith $N_1\\in\\mathscr{M}(\\tau)$ and $T\\in\\mathscr{A}(\\tau)$. Then $MN^*=M_1(S\nT^*)N_1^*$ and $S T^*\\in \\mathscr{A}(\\sigma\\wedge\\tau)$ so we may factorize it\nas $S T^*=UV^*$ with $U,V\\in \\mathscr{A}(\\sigma\\wedge\\tau)$, hence\n$MN^*=(M_1U)(N_1V)^*$.  By using \\eqref{eq:grm} we see that $M'=M_1U$\nand $N'=N_1V$ belong to $\\mathscr{M}(\\sigma\\wedge\\tau)$.  In particular, we\nhave $MN^*\\in\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$ if $M\\in\\mathscr{M}(\\sigma)$ and\n$N\\in\\mathscr{M}(\\tau)$.\n\nObserve that the assertion we just proved implies that\n$\\sum_\\sigma\\mathcal{K}(\\mathscr{M})(\\sigma)$ is dense in $\\mathcal{K}(\\mathscr{M})$.  \nIt remains to see that\n$\\mathcal{K}(\\mathscr{M})(\\sigma)\\mathcal{K}(\\mathscr{M})(\\tau)\\subset\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$.\nFor this it suffices that $M\\braket{M}{N}N^*$ be in\n$\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$ if $M\\in\\mathscr{M}(\\sigma)$ and\n$N\\in\\mathscr{M}(\\tau)$. Since $\\braket{M}{N}\\in\\mathscr{A}(\\sigma\\wedge\\tau)$ we\nmay write $\\braket{M}{N}=S T^*$ with $S,T\\in\\mathscr{A}(\\sigma\\wedge\\tau)$\nso $M\\braket{M}{N}N^*=(MS)(NT)^*\\in\\mathcal{K}(\\mathscr{M})(\\sigma\\wedge\\tau)$ by\n\\eqref{eq:grm}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip \n\n\nWe recall that the direct sum of a family $\\{\\mathscr{M}_i\\}$ of Hilbert\n$\\mathscr{A}$-modules is defined as follows: $\\oplus_i\\mathscr{M}_i$ is the space of\nelements $(M_i)_i\\in\\prod_i\\mathscr{M}_i$ such that the series\n$\\sum_i\\braket{M_i}{M_i}$ converges in $\\mathscr{A}$ equipped with the natural\n$\\mathscr{A}$-module structure and with the $\\mathscr{A}$-valued inner product defined\nby\n\\begin{equation}\\label{eq:sum}\n\\braket{(M_i)_i}{(N_i)_i} \n=\\textstyle\\sum_i\\braket{M_i}{N_i}.\n\\end{equation} \nThe algebraic direct sum of the $\\mathscr{A}$-modules $\\mathscr{M}_i$ is dense in\n$\\oplus_i\\mathscr{M}_i$.\n\n\nIt is easy to check that if each $\\mathscr{M}_i$ is graded and if we set\n$\\mathscr{M}(\\sigma)=\\oplus_i\\mathscr{M}_i(\\sigma)$ then $\\mathscr{M}$ becomes a graded\nHilbert $\\mathscr{A}$-module.  For example, if $\\mathscr{N}$ is a graded Hilbert\n$\\mathscr{A}$-module then $\\mathscr{N}\\oplus\\mathscr{A}$ is a graded Hilbert $\\mathscr{A}$-module and\nso the \\emph{linking algebra $\\mathcal{K}(\\mathscr{N}\\oplus\\mathscr{A})$ is equipped with a\ngraded algebra structure}. We recall \\cite[p. 50-52]{RW} that we\nhave a natural identification\n\\begin{equation}\\label{eq:link}\n\\mathcal{K}(\\mathscr{N}\\oplus\\mathscr{A})=\n\\begin{pmatrix}\n\\mathcal{K}(\\mathscr{N})& \\mathscr{N}\\\\\n\\mathscr{N}^*& \\mathscr{A}\n\\end{pmatrix}\n\\end{equation}\nand by Theorem \\ref{th:kghm} this is a graded algebra whose\n$\\sigma$-component is equal to\n\\begin{equation}\\label{eq:links}\n\\mathcal{K}(\\mathscr{N}(\\sigma)\\oplus\\mathscr{A}(\\sigma))=\n\\begin{pmatrix}\n\\mathcal{K}(\\mathscr{N}(\\sigma))& \\mathscr{N}(\\sigma)\\\\\n\\mathscr{N}(\\sigma)^*& \\mathscr{A}(\\sigma)\n\\end{pmatrix}.\n\\end{equation}\nIf $\\mathscr{N}$ is a $C^*$-submodule of $L(\\mathcal{E},\\mathcal{F})$ and if we set\n$\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{A},\\mathscr{N}\\cdot\\mathscr{N}^*=\\mathscr{B}$ then the linking algebra\n$\\begin{pmatrix}\\mathscr{B}& \\mathscr{M}\\\\\\mathscr{M}^*& \\mathscr{A}\\end{pmatrix}$ of $\\mathscr{M}$ is a\n$C^*$-algebra of operators on $\\mathcal{F}\\oplus\\mathcal{E}$.\n\nSome of the graded Hilbert $C^*$-modules which we shall use later on\nwill be constructed as follows.\n\n\\begin{proposition}\\label{pr:rhm}\nLet $\\mathcal{E},\\mathcal{F}$ be Hilbert spaces and let $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$ be a\nHilbert $C^*$-submodule, so that $\\mathscr{A}\\equiv\\mathscr{M}^*\\cdot\\mathscr{M}\\subset\nL(\\mathcal{E})$ is a $C^*$-algebra and $\\mathscr{M}$ is a full Hilbert\n$\\mathscr{A}$-module. Let $\\mathcal{C}$ be a $C^*$-algebra of operators on $\\mathcal{E}$\ngraded by the family of $C^*$-subalgebras\n$\\{\\mathcal{C}(\\sigma)\\}_{\\sigma\\in\\mathcal{S}}$. Assume that we have\n\\begin{equation}\\label{eq:tas}\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)=\\mathcal{C}(\\sigma)\\cdot\\mathscr{A}\n\\equiv\\mathscr{C}(\\sigma)\n\\hspace{2mm} \\text{for all } \\sigma\\in\\mathcal{S}\n\\end{equation}\nand that the family $\\{\\mathscr{C}(\\sigma)\\}$ of subspaces of $L(\\mathcal{F})$ is\nlinearly independent. Then the $\\mathscr{C}(\\sigma)$ are $C^*$-algebras of\noperators on $\\mathcal{E}$ and $\\mathscr{C}=\\sum^\\mathrm{c}_\\sigma\\mathscr{C}(\\sigma)$ is a\n$C^*$-algebra graded by the family $\\{\\mathscr{C}(\\sigma)\\}$. If\n$\\mathscr{N}(\\sigma)\\equiv\\mathscr{M}\\cdot\\mathcal{C}(\\sigma)$ then\n$\\mathscr{N}=\\sum^\\mathrm{c}_\\sigma\\mathscr{N}(\\sigma)$ is a full Hilbert $\\mathscr{C}$-module\ngraded by $\\{\\mathscr{N}(\\sigma)\\}$.\n\\end{proposition}\n\\proof\nWe have\n$$\n\\mathscr{C}(\\sigma)\\cdot\\mathscr{C}(\\tau)=\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\tau)\n=\\mathscr{A}\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathcal{C}(\\tau)\\subset\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\\mathscr{C}(\\sigma\\wedge\\tau).\n$$ \nThis proves that the $\\mathscr{C}(\\sigma)$ are $C^*$-algebras and that\n$\\mathscr{C}$ is $\\mathcal{S}$-graded. Then:\n$$\n\\mathscr{N}(\\sigma)\\cdot\\mathscr{C}(\\tau)=\\mathscr{M}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathcal{C}(\\tau)\\cdot\\mathscr{A}\n\\subset\\mathscr{M}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)\\cdot\\mathscr{A}=\n\\mathscr{M}\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\n\\mathscr{M}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\\mathscr{N}(\\sigma\\wedge\\tau)\n$$\nand\n$$\n\\mathscr{N}(\\sigma)^*\\cdot\\mathscr{N}(\\tau)=\n\\mathcal{C}(\\sigma)\\cdot\\mathscr{M}^*\\cdot\\mathscr{M}\\cdot\\mathcal{C}(\\tau)=\n\\mathcal{C}(\\sigma)\\cdot\\mathscr{A}\\cdot\\mathcal{C}(\\tau)=\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma)\\cdot\\mathcal{C}(\\tau)\\subset\n\\mathscr{A}\\cdot\\mathcal{C}(\\sigma\\wedge\\tau)=\\mathscr{C}(\\sigma\\wedge\\tau).\n$$\nObserve that this computation also gives\n$\\mathscr{N}(\\sigma)^*\\cdot\\mathscr{N}(\\sigma)=\\mathscr{C}(\\sigma)$.  Then\n$$\n\\big({\\textstyle\\sum_\\sigma}\\mathscr{N}(\\sigma)^*\\big)\n\\big({\\textstyle\\sum_\\sigma}\\mathscr{N}(\\sigma)\\big)=\n{\\textstyle\\sum_{\\sigma,\\tau}}\\mathscr{N}(\\sigma)^*\\mathscr{N}(\\tau)\\subset\n{\\textstyle\\sum_{\\sigma,\\tau}}\\mathscr{C}(\\sigma\\wedge\\tau)\\subset\n{\\textstyle\\sum_{\\sigma}}\\mathscr{C}(\\sigma)\n$$ and by the preceding remark we get $\\mathscr{N}^*\\cdot\\mathscr{N}=\\mathscr{C}$ so $\\mathscr{N}$\nis a full Hilbert $\\mathscr{C}$-module. To show the grading property it\nsuffices to prove that the family of subspaces $\\mathscr{N}(\\sigma)$ is\nlinearly independent. Assume that $\\sum N(\\sigma)=0$ with\n$N(\\sigma)\\in\\mathscr{N}(\\sigma)$ and $N(\\sigma)=0$ for all but a finite\nnumber of $\\sigma$. Assuming that there are non-zero elements in\nthis sum, let $\\tau$ be a maximal element of the set of $\\sigma$\nsuch that $N(\\sigma)\\neq0$. From\n$\\sum_{\\sigma_1,\\sigma_2}N(\\sigma_1)^*N(\\sigma_2)=0$ and since\n$N(\\sigma_1)^*N(\\sigma_2)\\in\\mathscr{C}(\\sigma_1\\wedge\\sigma_2)$ we get\n$\\sum_{\\sigma_1\\wedge\\sigma_2=\\sigma}N(\\sigma_1)^*N(\\sigma_2)=0$ for\neach $\\sigma$. Take here $\\sigma=\\tau$ and observe that if\n$\\sigma_1\\wedge\\sigma_2=\\tau$ and $\\sigma_1>\\tau$ or $\\sigma_2>\\tau$\nthen $N(\\sigma_1)^*N(\\sigma_2)=0$. Thus $N(\\tau)^*N(\\tau)=0$ so\n$N(\\tau)=0$. But this contradicts the choice of $\\tau$, so\n$N(\\sigma)=0$ for all $\\sigma$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Tensor products}\n\\label{ss:ha}\n\nIn this subsection we collect some facts concerning tensor products\nwhich are useful in what follows.  We recall the definition of the\ntensor product of a Hilbert space $\\mathcal{E}$ and a \\mbox{$C^*$-algebra}\n$\\mathscr{A}$ in the category of Hilbert $C^*$-modules, cf. \\cite{La}. We\nequip the algebraic tensor product $\\mathcal{E}\\odot\\mathscr{A}$ with the obvious\nright $\\mathscr{A}$-module structure and with the $\\mathscr{A}$-valued sesquilinear\nmap given by\n\\begin{equation}\\label{eq:hoa}\n\\braket{{\\textstyle\\sum_{u\\in\\mathcal{E}}}u\\otimes\n  A_u}{{\\textstyle\\sum_{v\\in\\mathcal{E}}}v\\otimes B_v} \n={\\textstyle\\sum_{u,v}}\\braket{u}{v}A^*_uB_v\n\\end{equation}\nwhere $A_u=B_u=0$ outside a finite set. Then the completion of\n$\\mathcal{E}\\odot\\mathscr{A}$ for the norm $\\|M\\|:=\\|\\braket{M}{M}\\|^{1\/2}$ is a\nfull Hilbert $\\mathscr{A}$-module denoted $\\mathcal{E}\\otimes\\mathscr{A}$. Clearly its\nimprimitivity algebra is\n\\begin{equation}\\label{eq:cha}\n\\mathcal{K}(\\mathcal{E}\\otimes\\mathscr{A})=K(\\mathcal{E})\\otimes\\mathscr{A}.\n\\end{equation}  \nIf $\\mathscr{A}$ is $\\mathcal{S}$-graded then $\\mathcal{E}\\otimes\\mathscr{A}$ is equipped with an\nobvious structure of $\\mathcal{S}$-graded Hilbert $\\mathscr{A}$-module.\n\nIf $\\mathscr{A}$ is realized on a Hilbert space $\\mathcal{F}$ then one has a natural\nisometric embedding $\\mathcal{E}\\otimes\\mathscr{A} \\subset L(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$.\nIndeed, there is a unique linear map $\\mathcal{E}\\otimes\\mathscr{A}\\to\nL(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$ which associates to $u\\otimes A$ the function\n$f\\mapsto u\\otimes(Af)$ and due to \\eqref{eq:hoa} this map is an\nisometry.  Thus the Hilbert $\\mathscr{A}$-module $\\mathcal{E}\\otimes\\mathscr{A}$ is realized\nas a Hilbert $C^*$-submodule of $L(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$, the dual\nmodule is realized as the set of adjoint operators\n$(\\mathcal{E}\\otimes\\mathscr{A})^*\\subset L(\\mathcal{E}\\otimes\\mathcal{F},\\mathcal{E})$, and one clearly has\n\\begin{equation}\\label{eq:etens}\n(\\mathcal{E}\\otimes\\mathscr{A})^*\\cdot (\\mathcal{E}\\otimes\\mathscr{A})=\\mathscr{A}, \\hspace{2mm}\n(\\mathcal{E}\\otimes\\mathscr{A})\\cdot(\\mathcal{E}\\otimes\\mathscr{A})^*=K(\\mathcal{E})\\otimes\\mathscr{A}.\n\\end{equation} \n\n\nIf $X$ is a locally compact space equipped with a Radon measure then\n$L^2(X)\\otimes\\mathscr{A}$ is the completion of $\\cc_{\\mathrm{c}}(X;\\mathscr{A})$ for the norm\n$\\|\\int_X F(x)^*F(x) \\text{d} x\\|^{1\/2}$.  Note that $L^2(X;\\mathscr{A})\\subset\nL^2(X)\\otimes\\mathscr{A}$ strictly in general, cf.\\ the example below.  If\n$\\mathscr{A}\\subset L(\\mathcal{F})$ then the norm on $L^2(X)\\otimes\\mathscr{A}$ is\n\\begin{equation}\\label{eq:L2a}\n\\|{\\textstyle\\int_X} F(x)^*F(x) \\text{d} x\\|^2=\n{\\textstyle\\sup_{f\\in\\mathcal{F},\\|f\\|=1}} {\\textstyle\\int_X} \n\\|F(x)f\\|^2 \\text{d} x.\n\\end{equation}\nIf $Y$ is a locally compact space then\n$\\mathcal{E}\\otimes\\cc_{\\mathrm{o}}(Y)\\cong\\cc_{\\mathrm{o}}(Y;\\mathcal{E})$.  Hence $L^2(X)\\otimes\\cc_{\\mathrm{o}}(Y)$ is\nthe completion of $\\cc_{\\mathrm{c}}(X\\times Y)$ for the norm $\\sup_{y\\in\n  Y}(\\int_X |F(x,y)|^2 \\text{d} x)^{1\/2}$. Assume that $X=Y$ is a\nlocally compact abelian group and let $f\\in L^\\infty(X)$ with\ncompact support and $g\\in L^2(X)$. It is easy to check that\n$F(x,y)=f(x)g(x+y)$ is an element of\n$\\cc_{\\mathrm{o}}(X;L^2(X))=L^2(X)\\otimes\\cc_{\\mathrm{o}}(X)$ but if $F(x,\\cdot)=f(x)U_x g$ is\nnot zero then it does not belong to $\\cc_{\\mathrm{o}}(X)$ and is not even a bounded\nfunction if $g$ is not. Thus the elements of $L^2(X)\\otimes\\mathscr{A}$ can\nnot be realized as bounded operator valued (equivalence classes of)\nfunctions on $X$.\n\n\n\nMore generally, if $\\mathcal{F}'$, $\\mathcal{F}''$ are Hilbert spaces and\n$\\mathscr{M}\\subset L(\\mathcal{F}',\\mathcal{F}'')$ is a closed subspace then we define\n$L^2(X)\\otimes\\mathscr{M}$ as the completion of the space $\\cc_{\\mathrm{c}}(X;\\mathscr{M})$ for a\nnorm similar to \\eqref{eq:L2a}. We clearly have\n$L^2(X)\\otimes\\mathscr{M}\\subset L(\\mathcal{F}',L^2(X)\\otimes\\mathcal{F}'')$ isometrically\nand $L^2(X;\\mathscr{M})\\subset L^2(X)\\otimes\\mathscr{M}$ continuously.\n\n\n\nIf $\\mathcal{E},\\mathcal{F},\\mathcal{G},\\mathcal{H}$ are Hilbert spaces and $\\mathscr{M}\\subset L(\\mathcal{E},\\mathcal{F})$\nand $\\mathscr{N}\\subset L(\\mathcal{G},\\mathcal{H})$ are closed linear subspaces then we\ndenote $\\mathscr{M}\\otimes\\mathscr{N}$ the closure in\n$L(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H})$ of the algebraic tensor product of\n$\\mathscr{M}$ and $\\mathscr{N}$. Now suppose that $\\mathscr{M}$ is a $C^*$-submodule of\n$L(\\mathcal{E},\\mathcal{F})$ and that $\\mathscr{N}$ is a $C^*$-submodule of $L(\\mathcal{G},\\mathcal{H})$ and\nlet $\\mathscr{A}=\\mathscr{M}^*\\cdot\\mathscr{M}$ and $\\mathscr{B}=\\mathscr{N}^*\\cdot\\mathscr{N}$. Then $\\mathscr{M}$ is a\nHilbert $\\mathscr{A}$-module and $\\mathscr{N}$ is a Hilbert $\\mathscr{B}$-module hence the\nexterior tensor product, denoted temporarily\n$\\mathscr{M}\\otimes_{\\text{ext}}\\mathscr{N}$, is well defined in the category of\nHilbert $C^*$-modules \\cite{La} and is a Hilbert\n$\\mathscr{A}\\otimes\\mathscr{B}$-module. On the other hand, it is easy to check that\n$(\\mathscr{M}\\otimes\\mathscr{N})^*=\\mathscr{M}^*\\otimes\\mathscr{N}^*$ and then that $\\mathscr{M}\\otimes\\mathscr{N}$\nis a Hilbert $C^*$-submodule of $L(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H})$\nsuch that\n$(\\mathscr{M}\\otimes\\mathscr{N})^*\\cdot(\\mathscr{M}\\otimes\\mathscr{N})=\\mathscr{A}\\otimes\\mathscr{B}$. Finally, it\nis clear that $L(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H})$ and\n$\\mathscr{M}\\otimes_{\\text{ext}}\\mathscr{N}$ induce the same $\\mathscr{A}\\otimes\\mathscr{B}$-valued\ninner product on the algebraic tensor product of $\\mathscr{M}$ and $\\mathscr{N}$.\nThus we we get a canonical isometric isomorphism\n$\\mathscr{M}\\otimes_{\\text{ext}}\\mathscr{N}=\\mathscr{M}\\otimes\\mathscr{N}$.\n\n\nAs an application we give now an abstract version of the \"toy\nmodels\" described in Example \\ref{ex:fried}. Let $\\mathcal{E},\\mathcal{F}$ be\nHilbert spaces and let us define $\\mathcal{H}=(\\mathcal{E}\\otimes\\mathcal{F})\\oplus\\mathcal{F}$.\nLet $\\mathscr{A}$ and $\\mathscr{B}$ be $C^*$-algebras of operators on $\\mathcal{F}$ and\n$\\mathcal{E}\\otimes\\mathcal{F}$ respectively. We embed $\\mathcal{E}\\otimes\\mathscr{A} \\subset\nL(\\mathcal{F},\\mathcal{E}\\otimes\\mathcal{F})$ as above. We simplify notation and denote\n$\\mathcal{E}^*\\otimes\\mathscr{A}:=(\\mathcal{E}\\otimes\\mathscr{A})^*\\subset L(\\mathcal{E}\\otimes\\mathcal{F},\\mathcal{F})$ the\ndual module.\n\n\\begin{proposition}\\label{pr:toymodel}\n  Let $\\mathcal{S}$ be a semilattice and $\\mathcal{T}$ an ideal of $\\mathcal{S}$. Assume\n  that the $C^*$-algebras $\\mathscr{A}$ and $\\mathscr{B}$ are $\\mathcal{S}$-graded and that\n  we have $\\mathscr{A}(\\sigma)=\\{0\\}$ if $\\sigma\\notin\\mathcal{T}$ and $\\mathscr{B}(\\tau)=\n  K(\\mathcal{E})\\otimes \\mathscr{A}(\\tau)$ for $\\tau\\in\\mathcal{T}$.  Then\n\\begin{equation}\\label{e:toy}\n\\mathscr{C}=\n\\begin{pmatrix}\n\\mathscr{B} & \\mathcal{E}\\otimes\\mathscr{A}\\\\\n\\mathcal{E}^*\\otimes\\mathscr{A} & \\mathscr{A}\n\\end{pmatrix}.\n\\end{equation}\nis an $\\mathcal{S}$-graded $C^*$-algebra if we define its components as\nfollows:\n\\begin{equation}\\label{e:gtoy}\n\\mathscr{C}(\\sigma)=\n\\begin{pmatrix}\n\\mathscr{B}(\\sigma) & \\mathcal{E}\\otimes\\mathscr{A}(\\sigma)\\\\\n\\mathcal{E}^*\\otimes\\mathscr{A}(\\sigma) & \\mathscr{A}(\\sigma)\n\\end{pmatrix} \\quad \\text{for all} \\quad \\sigma\\in\\mathcal{S}.\n\\end{equation}\n\\end{proposition}\n\\proof\nObserve that if we set $\\mathcal{T}'=\\mathcal{S}\\setminus\\mathcal{T}$ then\n\\begin{equation}\\label{e:imptoy}\n\\mathscr{C}=\n\\begin{pmatrix}\nK(\\mathcal{E})\\otimes\\mathscr{A} & \\mathcal{E}\\otimes\\mathscr{A}\\\\\n\\mathcal{E}^*\\otimes\\mathscr{A} & \\mathscr{A}\n\\end{pmatrix} +\n\\begin{pmatrix}\n\\mathscr{B}(\\mathcal{T}') &  0 \\\\\n0 & 0\n\\end{pmatrix} =\n\\mathcal{K}(\\mathscr{N}\\oplus\\mathscr{A}) +\n\\begin{pmatrix}\n\\mathscr{B}(\\mathcal{T}') &  0 \\\\\n0 & 0\n\\end{pmatrix}\n\\end{equation}\nwhere $\\mathscr{N}=\\mathcal{E}\\otimes\\mathscr{A}$ is an $\\mathcal{S}$-graded Hilbert $\\mathscr{A}$-module,\ncf. \\eqref{eq:link} and \\eqref{eq:cha}. It is easy to see that the\nfamily $\\{\\mathscr{C}(\\sigma)\\}$ is linearly independent and that $\\mathscr{C}$ is\nthe closure of its sum. By taking into account \\eqref{eq:links} we\nsee that it suffices to show that\n$\\mathscr{C}(\\sigma)\\mathscr{C}(\\tau)\\subset\\mathscr{C}(\\sigma\\wedge\\tau)$ if\n$\\sigma\\in\\mathcal{T}'$ and $\\tau\\in\\mathcal{T}$. After computing the coefficients\nof the matrices we see that it suffices to check that\n$\\mathscr{B}(\\sigma)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau)\n\\subset\\mathcal{E}\\otimes\\mathscr{A}(\\sigma\\wedge\\tau)$. \nBut:\n\\begin{align*}\n\\mathscr{B}(\\sigma)\\cdot\\mathcal{E}\\otimes\\mathscr{A}(\\tau) & =\n\\mathscr{B}(\\sigma)\\cdot K(\\mathcal{E})\\otimes\\mathscr{A}(\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau) = \n\\mathscr{B}(\\sigma)\\cdot\\mathscr{B}(\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau) \\\\\n& \\subset \\mathscr{B}(\\sigma\\wedge\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau)\n=K(\\mathcal{E})\\otimes\\mathscr{A}(\\sigma\\wedge\\tau)\\cdot \\mathcal{E}\\otimes\\mathscr{A}(\\tau)\n\\subset \\mathcal{E}\\otimes\\mathscr{A}(\\sigma\\wedge\\tau)\n\\end{align*}\nwhich finishes the proof.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nThe extension to an increasing family of ideals\n$\\mathcal{T}_1\\subset\\mathcal{T}_2\\dots\\subset\\mathcal{S}$ is straightforward. \n\n\n\\section{The many-body $C^*$-algebra }\n\\label{s:grass}\n\\protect\\setcounter{equation}{0}\n\nIn this section we introduce the many-body $C^*$-algebra and\ndescribe its main properties (in particular, we prove the theorems\n\\ref{th:C} and \\ref{th:CG}). Subsection \\ref{ss:hilbert} contains\nsome preparatory material on concrete realizations of Hilbert\n$C^*$-modules which implement the Morita equivalence between some\ncrossed products.\n\n\n\\subsection{Notations}\n\\label{ss:group}\n\n\nLet $X$ be a locally compact abelian group with operation denoted\nadditively equipped with a Haar measures $\\text{d} x$. We abbreviate\nthis by saying that \\emph{$X$ is an lca group}.  We set $\\mathscr{L}_X\\equiv\nL(L^2(X))$ and $\\mathscr{K}_X\\equiv K(L^2(X))$ and note that these are\n$C^*$-algebras independent of the choice of the measure on $X$.  If\n$Y$ is a second lca group we shall use the abbreviations\n\\begin{equation}\\label{eq:lkxy}\n\\mathscr{L}_{XY}=L(L^2(Y),L^2(X)) \\quad\\text{and}\\quad\n\\mathscr{K}_{XY}=K(L^2(Y),L^2(X)). \n\\end{equation}\nWe denote by $\\varphi(Q)$ the operator in $L^2(X)$ of multiplication\nby a function $\\varphi$ and if $X$ has to be explicitly specified we\nset $Q=Q_X$.  The bounded uniformly continuous functions on $X$ form\na $C^*$-algebra $\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ which contains the algebras $\\cc_{\\mathrm{c}}(X)$ and\n$\\cc_{\\mathrm{o}}(X)$.  The map $\\varphi\\mapsto\\varphi(Q)$ is an embedding\n$\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)\\subset\\mathscr{L}_X$.\n\nThe group $\\mathcal{C}^*$-algebra $\\mathscr{T}_X$ of $X$ is the closed linear\nsubspace of $\\mathscr{L}_X$ generated by the convolution operators of the\nform $(\\varphi*f)(x)=\\int_X \\varphi(x-y)f(y)\\text{d} y$ with\n$\\varphi\\in\\cc_{\\mathrm{c}}(X)$.  Observe that $f\\mapsto\\varphi*f$ is equal to\n$\\int_X\\varphi(-a)U_a\\,\\text{d} a$ where $U_a$ is the unitary\ntranslation operator on $L^2(X)$ defined by $(U_af)(x)=f(x+a)$.\n\nLet $X^*$ be the group dual to $X$ with operation denoted\nadditively\\symbolfootnote[2]{\\ Then $(k+p)(x)=k(x)p(x)$, $0(x)=1$,\n  and the element $-k$ of $X^*$ represents the function\n  $\\bar{k}$. In order to avoid such strange looking expressions one\n  might use the notation $k(x)=[x,k]$.  }.  If $k\\in X^*$ we define\na unitary operator $V_k$ on $L^2(X)$ by $(V_ku)(x)=k(x) u(x)$.  The\nFourier transform of an integrable measure $\\mu$ on $X$ is defined\nby $(F\\mu)(k)=\\int \\bar{k}(x)\\mu(\\text{d} x)$.  Then $F$ induces a\nbijective map $L^2(X)\\rightarrow L^2(X^*)$ hence a canonical isomorphism\n$S\\mapsto F^{-1}S F$ of $\\mathscr{L}_{X^*}$ onto $\\mathscr{L}_X$.  If $\\psi$ is a\nfunction on $X^*$ we set $\\psi(P)\\equiv\\psi(P_X)=F^{-1}M_\\psi F$,\nwhere $M_\\psi=\\psi(Q_{X^*})$ is the operator of multiplication by\n$\\psi$ on $L^2(X^*)$.  The map $\\psi\\mapsto\\psi(P)$ gives an\nisomorphism $\\cc_{\\mathrm{o}}(X^*)\\cong \\mathscr{T}_X$.\n\n\nIf $Y\\subset X$ is a closed subgroup then $\\pi_Y:X\\to X\/Y$ is the\ncanonical surjection.  We embed $\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ with the\nhelp of the injective morphism $\\varphi\\mapsto\\varphi\\circ\\pi_Y$. So\n$\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)$ is identified with the set of functions\n$\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ such that $\\varphi(x+y)=\\varphi(x)$ for all\n$x\\in X$ and $y\\in Y$.\n\nIn particular, $\\cc_{\\mathrm{o}}(X\/Y)$ is identified with the set of continuous\nfunctions $\\varphi$ on $X$ such that $\\varphi(x+y)=\\varphi(x)$ for\nall $x\\in X$ and $y\\in Y$ and such that for each $\\varepsilon>0$\nthere is a compact $K\\subset X$ such that $|\\varphi(x)|<\\varepsilon$\nif $x\\notin K+Y$. By $x\/Y\\to\\infty$ we mean $\\pi_Y(x)\\to\\infty$, so\nthe last condition is equivalent to $\\varphi(x)\\to0$ if\n$x\/Y\\to\\infty$.  For coherence with later notations we set\n\\begin{equation}\\label{eq:coxy}\n\\mathcal{C}_X(Y)=\\cc_{\\mathrm{o}}(X\/Y)\n\\end{equation}\nObserve that to an element $y\\in Y$ we may associate a translation\noperator $U_y$ in $L^2(X)$ and another translation operator in\n$L^2(Y)$. However, in order not to overcharge the writing we shall\ndenote the second operator also by $U_y$.  The restriction map\n$k\\mapsto k|_Y$ is a continuous surjective group morphism $X^*\\to\nY^*$ with kernel equal to $Y^\\perp=\\{k\\in X^*\\mid\nk(y)=1\\hspace{1mm}\\forall y\\in Y\\}$ which defines the canonical\nidentification $Y^*\\cong X^*\/Y^\\perp$. We denote by the same symbol\n$V_k$ the operator of multiplication by the character $k\\in X^*$ in\n$L^2(X)$ and by the character $k|_Y\\in Y^*$ in $L^2(Y)$.\n\nWe shall write $X=Y\\oplus Z$ if $X$ is the direct sum of the two\nclosed subgroups $Y,Z$ equipped with compatible Haar measures, in\nthe sense that $\\text{d} x=\\text{d} y\\otimes \\text{d} z$.  Then\n$L^2(X)=L^2(Y)\\otimes L^2(Z)$ as Hilbert spaces and\n$\\mathscr{K}_X=\\mathscr{K}_Y\\otimes \\mathscr{K}_Z$ and $\\mathcal{C}_X(Y)=1\\otimes\\cc_{\\mathrm{o}}(Z)$ as\n$C^*$-algebras.\n\nLet $O=\\{0\\}$ be the trivial group equipped with the Haar measure of\ntotal mass $1$. Then $L^2(O)=\\mathbb{C}$.\n\n\\subsection{Crossed products}\n\\label{ss:nbcrp}\n\nLet $X$ be a locally compact abelian group. A $C^*$-subalgebra\n$\\mathcal{A}\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ stable under translations will be called\n\\emph{$X$-algebra}.  The \\emph{crossed product of\n  $\\mathcal{A}$ by the action of $X$} is an abstractly defined $C^*$-algebra\n$\\mathcal{A}\\rtimes X$ canonically identified with the $C^*$-algebra of\noperators on $L^2(X)$ given by\n\\begin{equation}\\label{eq:crp}\n\\mathcal{A}\\rtimes X\\equiv\\mathcal{A}\\cdot \\mathscr{T}_X=\\mathscr{T}_X\\cdot\\mathcal{A}.\n\\end{equation}\nCrossed products of the form $\\mathcal{C}_X(Y)\\rtimes X$ where $Y$ is a\nclosed subgroup of $X$ play an important role in the many-body\nproblem. To simplify notations we set\n\\begin{equation}\\label{eq:Cxy}\n\\mathscr{C}_X(Y)=\\mathcal{C}_X(Y)\\rtimes X=\\mathcal{C}_X(Y)\\cdot\\mathscr{T}_X=\n\\mathscr{T}_X\\cdot\\mathcal{C}_X(Y).\n\\end{equation}\nIf $X=Y\\oplus Z$ and if we identify $L^2(X)=L^2(Y)\\otimes L^2(Z)$ then\n$\\mathscr{T}_X=\\mathscr{T}_Y\\otimes\\mathscr{T}_Z$ hence\n\\begin{equation}\\label{eq:crxyz}\n\\mathscr{C}_X(Y)=\\mathscr{T}_Y\\otimes \\mathscr{K}_Z.\n\\end{equation}\nA useful ``symmetric'' description of $\\mathscr{C}_X(Y)$ is contained in the\nnext lemma. Let $Y^{(2)}$ be the closed subgroup of $X^2\\equiv X\\oplus\nX$ consisting of elements of the form $(y,y)$ with $y\\in Y$.\n\n\\begin{lemma}\\label{lm:sym}\n$\\mathscr{C}_X(Y)$ is the closure of the\nset of integral operators with kernels $\\theta\\in\\cc_{\\mathrm{c}}(X^2\/Y^{(2)})$.\n\\end{lemma}\n\\proof Let $\\mathscr{C}$ be the norm closure of the set of integral\noperators with kernels $\\theta\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X^2)$ having the properties:\n(1) $\\theta(x+y,x'+y)=\\theta(x,x')$ for all $x,x'\\in X$ and $y\\in\nY$; (2) $\\mbox{\\rm supp\\! }\\theta\\subset K_\\theta+Y$ for some compact\n$K_\\theta\\subset X^2$. We show $\\mathscr{C}=\\mathscr{C}_X(Y)$.  Observe that the map\nin $X^2$ defined by $(x,x')\\mapsto(x-x',x')$ is a topological group\nisomorphism with inverse $(x_1,x_2)\\mapsto(x_1+x_2,x_2)$ and sends\nthe subgroup $Y^{(2)}$ onto the subgroup $\\{0\\}\\oplus Y$. This map\ninduces an isomorphism $X^2\/Y^{(2)}\\simeq X\\oplus(X\/Y)$. Thus any\n$\\theta\\in\\cc_{\\mathrm{c}}(X^2\/Y^{(2)})$ is of the form\n$\\theta(x,x')=\\widetilde\\theta(x-x',x')$ for some\n$\\widetilde\\theta\\in\\cc_{\\mathrm{c}}(X\\oplus(X\/Y))$. Thus $\\mathscr{C}$ is the closure in\n$\\mathscr{L}_X$ of the set of operators of the form\n$(Tu)(x)=\\int_X\\widetilde\\theta(x-x',x')u(x') \\text{d} x'$. Since we may\napproximate $\\widetilde\\theta$ with linear combinations of functions of\nthe form $a\\otimes b$ with $a\\in\\cc_{\\mathrm{c}}(X), b\\in\\cc_{\\mathrm{c}}(X\/Y)$ we see that\n$\\mathscr{C}$ is the clspan of the set of operators of the form\n$(Tu)(x)=\\int_X a(x-x')b(x')u(x') \\text{d} x'$. But this clspan is\n$\\mathscr{T}_X\\cdot\\mathcal{C}_X(Y)=\\mathscr{C}_X(Y)$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Compatible subgroups}\n\\label{ss:compat}\n\nIf $X,Y$ is an arbitrary pair of lca groups then $X\\oplus Y$ is the\nset $X\\times Y$ equipped with the product topology and group\nstructure. If $X,Y$ are closed subgroups of an lca group $G$ and if\nthe map $Y\\oplus Z\\to Y+Z$ defined by $(y,z)\\mapsto y+z$ is open, we\nsay that they are \\emph{compatible subgroups of $G$}.  In this case\n$Y+Z$ is a closed subgroup of $X$.\n\n\n\\begin{remark}\\label{re:DP}{\\rm\nIf $G$ is $\\sigma$-compact then $X,Y$ are compatible if and only if\n$X+Y$ is closed. Indeed, a continuous surjective morphism between\ntwo locally compact $\\sigma$-compact groups is open and a subgroup\n$H$ of a locally compact group $G$ is closed if and only if $H$ is\nlocally compact for the induced topology, see Theorems 5.11 and 5.29\nin \\cite{HR}.  We thank Lo\\\"ic Dubois and Benoit Pausader for\nenlightening discussions on this matter.\n}\\end{remark}\n\n\nThe importance of the compatibility condition in the context of\ngraded $C^*$-algebras has been pointed out in \\cite[Lemma 6.1.1]{Ma}\nand one may find there several descriptions of this condition (see\nalso Lemma 3.1 from \\cite{Ma3}). We quote two of them. Let $X\/Y$ be\nthe image of $X$ in $G\/Y$ considered as a subgroup of $G\/Y$ equipped\nwith the induced topology. The group $X\/(X\\cap Y)$ is equipped with\nthe locally compact quotient topology and we have a natural map\n$X\/(X\\cap Y)\\to X\/Y$ which is a bijective continuous group morphism.\nThen $X,Y$ are compatible if and only if the following equivalent\nconditions are satisfied:\n\\begin{align}\n& \\text{the natural map} \\hspace{2mm} X\/(X\\cap Y)\\to X\/Y \\hspace{2mm}\n\\text{is a homeomorphism}, \\label{eq:ma1} \\\\\n&\n\\text{the natural map }\nG\/(X\\cap Y)\\to G\/X\\times G\/Y \\hspace{1mm} \\text{is closed}.\n\\label{eq:ma2}\n\\end{align}\n\nIf $\\mathcal{A}$ is a $G$-algebra let $\\mathcal{A}|_X$ be the set of restrictions\nto $X$ of the functions from $\\mathcal{A}$. This is an $X$-algebra.\n\n\\begin{lemma}\\label{lm:reg}\nIf $X,Y$ are compatible subgroups of $G$ then\n\\begin{align}\n& \\mathcal{C}_G(X)\\cdot\\mathcal{C}_G(Y) = \\mathcal{C}_G(X\\cap Y)  \\label{eq:reg1}\\\\\n& \\mathcal{C}_G(Y)|_X =  \\mathcal{C}_X(X\\cap Y).\n\\label{eq:reg2}\n\\end{align}\nThe second relation remains valid for the subalgebras $\\cc_{\\mathrm{c}}$.\n\\end{lemma}\n\\proof The fact that the inclusion $\\subset$ in \\eqref{eq:reg1} is\nequivalent to the compatibility of $X$ and $Y$ is shown in Lemma\n6.1.1 from \\cite{Ma}, so we only have to prove that the equality\nholds. Let $E = (G\/X) \\times (G\/Y)$. If $\\varphi \\in \\cc_{\\mathrm{o}}(G\/X)$ and\n$\\psi \\in \\cc_{\\mathrm{o}}(G\/Y)$ then $\\varphi \\otimes \\psi$ denotes the function\n$(s, t) \\longmapsto \\varphi(s) \\psi(t)$, which belongs to\n$\\cc_{\\mathrm{o}}(E)$. The subspace generated by the functions of the form\n$\\varphi \\otimes \\psi$ is dense in $\\cc_{\\mathrm{o}}(E)$ by the Stone-Weierstrass\ntheorem. If $F$ is a closed subset of $E$ then, by the Tietze\nextension theorem, each function in $\\cc_{\\mathrm{c}}(F)$ extends to a function\nin $\\cc_{\\mathrm{c}}(E)$, so the restrictions $(\\varphi \\otimes \\psi)|_F$\ngenerate a dense linear subspace of $\\cc_{\\mathrm{o}}(F)$.  Let us denote by\n$\\pi$ the map $x \\mapsto (\\pi_X(x), \\pi_Y(x))$, so $\\pi$ is a group\nmorphism from $G$ to $E$ with kernel $V=X\\cap Y$.  Then by\n\\eqref{eq:ma2} the range $F$ of $\\pi$ is closed and the quotient map\n$\\widetilde\\pi : G\/V \\to F$ is a continuous and closed bijection, hence\nis a homeomorphism. So $\\theta \\mapsto \\theta \\circ \\tilde \\pi$ is\nan isometric isomorphism of $\\cc_{\\mathrm{o}}(F)$ onto $\\cc_{\\mathrm{o}}(G\/V)$. Hence for\n$\\varphi \\in \\cc_{\\mathrm{o}}(G\/X)$ and $\\psi \\in \\cc_{\\mathrm{o}}(G\/Y)$ the function $\\theta\n= (\\varphi \\otimes \\psi) \\circ \\tilde \\pi$ belongs to $\\cc_{\\mathrm{o}}(G\/V)$, it\nhas the property $\\theta \\circ \\pi_V = \\varphi \\circ \\pi_X \\cdot\n\\psi \\circ \\pi_Y$, and the functions of this form generate a dense\nlinear subspace of $\\cc_{\\mathrm{o}}(G\/V)$.\n\nNow we prove \\eqref{eq:reg2}. Recall that we identify $\\mathcal{C}_G(Y)$\nwith a subset of $\\cc_{\\mathrm{b}}^{\\mathrm{u}}(G)$ by using $\\varphi\\mapsto\\varphi\\circ\\pi_Y$\nso in terms of $\\varphi$ the restriction map which defines\n$\\mathcal{C}_G(Y)|_X$ is just $\\varphi\\mapsto\\varphi|_{X\/Y}$. Thus we have a\ncanonical embedding $\\mathcal{C}_G(Y)|_X\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)$ for an arbitrary\npair $X,Y$ . Then the continuous bijective group morphism\n$\\theta:X\/(X\\cap Y)\\to X\/Y$ allows us to embed\n$\\mathcal{C}_G(Y)|_X\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/(X\\cap Y))$. That the range of this map is\nnot $\\mathcal{C}_X(X\\cap Y)$ in general is clear from the example $G=\\mathbb{R},\nX=\\pi\\mathbb{Z},Y=\\mathbb{Z}$. But if $X,Y$ are compatible then $X\/Y$ is closed\nin $G\/Y$, so $\\mathcal{C}_G(Y)|_X=\\cc_{\\mathrm{o}}(X\/Y)$ by the Tietze extension theorem,\nand $\\theta$ is a homeomorphism, hence we get \\eqref{eq:reg2}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\\begin{lemma}\\label{lm:double}\nIf $X,Y$ are compatible subgroups of $G$ then $X^2=X\\oplus X$ and\n$Y^{(2)}=\\{(y,y)\\mid y\\in Y\\}$ is a compatible pair of closed\nsubgroups of $G^2=G\\oplus G$.\n\\end{lemma}\n\\proof Let $D=X^2\\cap Y^{(2)}=\\{(x,x)\\mid x\\in X\\cap Y\\}$. Due to to\n\\eqref{eq:ma1} it suffices to show that the natural map\n$Y^{(2)}\/D\\to Y^{(2)}\/X^2$ is a homeomorphism.  Here $Y^{(2)}\/X^2$\nis the image of $Y^{(2)}$ in $G^2\/X^2\\cong (G\/X)\\oplus(G\/X)$, more\nprecisely it is the subset of pairs $(a,a)$ with $a=\\pi_X(z)$ and\n$z\\in Y$, equipped with the topology induced by\n$(G\/X)\\oplus(G\/X)$. Thus the natural map $Y\/X\\to Y^{(2)}\/X^2$ is a\nhomeomorphism. On the other hand, the natural map $Y\/(X\\cap Y)\\to\nY^{(2)}\/D$ is clearly a homeomorphism.  To finish the proof note\nthat $Y\/(X\\cap Y)\\to Y\/X$ is a homeomorphism because $X,Y$ is a\nregular pair.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{lemma}\\label{lm:regp}\n  Let $X,Y$ be compatible subgroups of an lca group $G$ and let\n  $X^\\perp,Y^\\perp$ be their orthogonals in $G^*$.  Then $(X\\cap\n  Y)^\\perp=X^\\perp+Y^\\perp$ and the closed subgroups\n  $X^\\perp,Y^\\perp$ of $G^*$ are compatible.\n\\end{lemma}\n\\proof $X+Y$ is closed and, since\n$(x,y)\\mapsto(x,-y)$ is a homeomorphism, the map $S:X\\oplus Y\\to\nX+Y$ defined by $S(x,y)=x+y$ is an open surjective morphism. Then\nfrom the Theorem 9.5, Chapter 2 of \\cite{Gu} it follows that the\nadjoint map $S^*$ is a homeomorphism between $(X+Y)^*$ and its\nrange. In particular its range is a locally compact subgroup for the\ntopology induced by $X^*\\oplus Y^*$ hence is a closed subgroup of\n$X^*\\oplus Y^*$, see Remark \\ref{re:DP}.  We\nhave $(X+Y)^\\perp=X^\\perp\\cap Y^\\perp$, cf. 23.29 in \\cite{HR}. Thus\nfrom $X^*\\cong G^*\/X^\\perp$ and similar representations for $Y^*$\nand $(X+Y)^*$ we see that\n$$\nS^*:G^*\/(X^\\perp \\cap Y^\\perp)\\to G^*\/X^\\perp\\oplus G^*\/Y^\\perp\n$$ is a closed map. But $S^*$ is clearly the natural map involved in\n\\eqref{eq:ma2}, hence the pair $X^\\perp,Y^\\perp$ is\nregular. Finally, note that $(X\\cap Y)^\\perp$ is always equal to the\nclosure of the subgroup $X^\\perp+Y^\\perp$, cf. 23.29 and 24.10 in\n\\cite{HR}, and in our case $X^\\perp+Y^\\perp$ is closed.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Green Hilbert $C^*$-modules} \n\\label{ss:hilbert} \n\nLet $X,Y$ be a compatible pair of closed subgroups of a locally\ncompact abelian group $G$. Then the subgroup $X+Y$ of $G$ generated\nby $X\\cup Y$ is also closed. If we identify $X\\cap Y$ with the\nclosed subgroup $D$ of $X\\oplus Y$ consisting of the elements of the\nform $(z,z)$ with $z\\in X\\cap Y$ then the quotient group $X\\uplus Y\n\\equiv (X\\oplus Y)\/(X\\cap Y)$ is locally compact and the map\n\\begin{equation}\\label{eq:nat}\n\\phi:X\\oplus Y \\to X+Y \\hspace{2mm}\\text{defined by}\\hspace{2mm}\n \\phi(x,y)=x-y\n\\end{equation}\nis an open continuous surjective group morphism $X\\oplus Y\\to X+Y$\nwith $X\\cap Y$ as kernel.  Hence the group morphism\n$\\phi^\\circ:X\\uplus Y\\to X+Y$ induced by $\\phi$ is a homeomorphism.\n\n\nSince $\\cc_{\\mathrm{c}}(X\\uplus Y)\\subset \\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\\oplus Y)$ the elements\n$\\theta\\in\\cc_{\\mathrm{c}}(X\\uplus Y)$ are functions $\\theta:X\\times Y\\to\\mathbb{C}$\nand we may think of them as kernels of integral operators.\n\n\\begin{lemma}\\label{lm:bound}\n  If $\\theta\\in\\cc_{\\mathrm{c}}(X\\uplus Y)$ then $(T_\\theta\n  u)(x)=\\int_Y\\theta(x,y)u(y) \\text{d} y$ defines an operator in\n  $\\mathscr{L}_{XY}$ with norm $\\|T_\\theta\\|\\leq C\\sup|\\theta|$ where $C$\n  depends only on a compact which contains the support of $\\theta$.\n\\end{lemma}\n\\proof \nBy the Schur test\n$$\n\\|T_\\theta\\|^2\\leq \n{\\textstyle\\sup_{x\\in X}}\\int_Y|\\theta(x,y)\\text{d}  y \\cdot\n{\\textstyle\\sup_{y\\in Y}}\\int_X|\\theta(x,y)\\text{d} x.\n$$ \nLet $K\\subset X$ and $L\\subset Y$ be compact sets such that\n$(K\\times L) + D$ contains the support of $\\theta$.\nThus if $\\theta(x,y)\\neq0$ then $x\\in z+K$ and $y\\in z+L$ for some \n$z\\in X\\cap Y$ hence $ \\int_Y|\\theta(x,y)\\text{d} y \\leq\n\\sup|\\theta| \\lambda_Y(L).  $ Similarly $\\int_X|\\theta(x,y)\\text{d} x\n\\leq \\sup|\\theta| \\lambda_X(K)$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{definition}\\label{df:ryz}\n$\\mathscr{T}_{XY}$ is the norm closure in $\\mathscr{L}_{XY}$ of the set of operators\n$T_\\theta$ as in Lemma \\ref{lm:bound}.\n\\end{definition}\n\n\n\n\\begin{remark}\\label{re:rief}\n  If $X\\supset Y$ then $\\mathscr{T}_{XY}$ is a ``concrete'' realization of\n  the Hilbert $C^*$-module introduced by Rieffel in \\cite{Ri} which\n  implements the Morita equivalence between the group $C^*$-algebra\n  $\\mathcal{C}^*(Y)$ and the crossed product $\\cc_{\\mathrm{o}}(X\/Y)\\rtimes X$. More\n  precisely, $\\mathscr{T}_{XY}$ is a Hilbert $\\mathcal{C}^*(Y)$-module and its\n  imprimitivity algebra is canonically isomorphic with\n  $\\cc_{\\mathrm{o}}(X\/Y)\\rtimes X$.  If $X,Y$ is an arbitrary couple of\n  compatible subgroups of $G$ then we defined $\\mathscr{T}_{XY}$ such that\n  $\\mathscr{T}_{XY}=\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{GY}$. On the other hand, from\n  \\eqref{eq:factor} we get $\\mathscr{T}_{XY}=\\mathscr{T}_{XE}\\cdot\\mathscr{T}_{EY}$ with\n  $E=X\\cap Y$, hence $\\mathscr{T}_{XY}$ is naturally a Hilbert\n  $(\\cc_{\\mathrm{o}}(X\/E)\\rtimes X,\\cc_{\\mathrm{o}}(Y\/E)\\rtimes Y)$ imprimitivity bimodule.\n  It has been noticed by Georges Skandalis that $\\mathscr{T}_{XY}$ is in\n  fact a ``concrete'' realization of a Hilbert $C^*$-module\n  introduced by Green to show the Morita equivalence of the\n  $C^*$-algebras $\\cc_{\\mathrm{o}}(Z\/Y)\\rtimes X$ and $\\cc_{\\mathrm{o}}(Z\/X)\\rtimes Y$ where\n  we take $Z=X+Y$, cf. \\cite[Example 4.13]{Wi}.\n\\end{remark}\n\n\nWe give now an alternative definition of $\\mathscr{T}_{XY}$. If\n$\\varphi\\in\\cc_{\\mathrm{c}}(G)$ we define $T_{XY}(\\varphi):\\cc_{\\mathrm{c}}(Y)\\to\\cc_{\\mathrm{c}}(X)$ by\n\\begin{equation}\\label{eq:ryz}\n(T_{XY}(\\varphi)u)(x)=\\int_Y\\varphi(x-y)u(y)\\text{d} y.\n\\end{equation}\nThis operator depends only the restriction $\\varphi|_{X+Y}$ hence,\nby the Tietze extension theorem, we could take $\\varphi\\in\\cc_{\\mathrm{c}}(Z)$\ninstead of $\\varphi\\in\\cc_{\\mathrm{c}}(G)$, where $Z$ is any closed subgroup of\n$G$ containing $X\\cup Y$.\n\n\\begin{proposition}\\label{pr:def2}\n$T_{XY}(\\varphi)$ extends to a bounded operator $L^2(Y)\\to L^2(X)$,\nalso denoted $T_{XY}(\\varphi)$, and for each compact $K\\subset G$\nthere is a constant $C$ such that if $\\mbox{\\rm supp\\! }\\varphi\\subset K$\n\\begin{equation}\\label{eq:nyz}\n\\|T_{XY}(\\varphi)\\|\\leq C \\sup\\nolimits_{x\\in G}|\\varphi(x)|.\n\\end{equation}\nThe adjoint operator is given by $T_{XY}(\\varphi)^*=\nT_{YX}(\\varphi^*)$ where $\\varphi^*(x)=\\bar\\varphi(-x)$.  The space\n$\\mathscr{T}_{XY}$ coincides with the closure in $\\mathscr{L}_{XY}$ of the set of\noperators of the from $T_{XY}(\\varphi)$.\n\\end{proposition}\n\\proof The set $X+Y$ is closed in $G$ hence the restriction map\n$\\cc_{\\mathrm{c}}(G)\\to\\cc_{\\mathrm{c}}(X+Y)$ is surjective. On the other hand, the map\n$\\phi^\\circ:X\\uplus Y\\to X+Y$, defined after \\eqref{eq:nat}, is a\nhomeomorphism so it induces an isomorphism\n$\\varphi\\to\\varphi\\circ\\phi^\\circ$ of $\\cc_{\\mathrm{c}}(X+Y)$ onto $\\cc_{\\mathrm{c}}(X\\uplus\nY)$. Clearly $T_{XY}(\\varphi)=T_\\theta$ if $\\theta=\\varphi\\circ\\phi$,\nso the proposition follows from Lemma \\ref{lm:bound}.  \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nWe discuss now some properties of the spaces $\\mathscr{T}_{XY}$.\nWe set $\\mathscr{T}_{XY}^*\\equiv(\\mathscr{T}_{XY})^*\\subset\\mathscr{L}_{YX}$. \n\n\\begin{proposition}\\label{pr:nyza}\nWe have  $\\mathscr{T}_{XX}=\\mathscr{T}_X$ and:\n\\begin{align}\n& \\mathscr{T}_{XY}^* =\\mathscr{T}_{YX} \\label{eq:rad} \\\\\n& \\mathscr{T}_{XY}   =\\mathscr{T}_{XY}\\cdot \\mathscr{T}_Y=\\mathscr{T}_X\\cdot\\mathscr{T}_{XY}\n\\label{eq:cyzc}  \\\\\n& \\mathcal{A}|_X\\cdot\\mathscr{T}_{XY} =\\mathscr{T}_{XY}\\cdot\\mathcal{A}|_Y \\label{eq:ayza}\n\\end{align}\nwhere $\\mathcal{A}$ is an arbitrary  $G$-algebra.\n\\end{proposition}\n\\proof The relations $\\mathscr{T}_{XX}=\\mathscr{T}_X$ and \\eqref{eq:rad} are\nobvious.  Now we prove the first equality in \\eqref{eq:cyzc} (then\nthe second one follows by taking adjoints). If $C(\\eta)$ is the\noperator of convolution in $L^2(Y)$ with $\\eta\\in \\cc_{\\mathrm{c}}(Y)$ then a\nshort computation gives\n\\begin{equation}\\label{eq:yzc}\nT_{XY}(\\varphi)C(\\eta)=T_{XY}(T_{G Y}(\\varphi)\\eta)\n\\end{equation}\nfor $\\varphi\\in\\cc_{\\mathrm{c}}(G)$. Since $T_{G Y}(\\varphi)\\eta\\in\\cc_{\\mathrm{c}}(G)$ we get\n$T_{XY}(\\varphi)C(\\eta)\\in\\mathscr{T}_{GX}$, so $\\mathscr{T}_{XY}\\cdot\n\\mathscr{T}_Y\\subset\\mathscr{T}_{XY}$. The converse follows by a standard\napproximation argument.\n\nLet $\\varphi\\in\\cc_{\\mathrm{c}}(G)$ and $\\theta\\in\\mathcal{A}$. We shall denote by\n$\\theta(Q_X)$ the operator of multiplication by $\\theta|_X$ in\n$L^2(X)$ and by $\\theta(Q_Y)$ that of multiplication by $\\theta|_Y$\nin $L^2(Y)$.  Choose some $\\varepsilon>0$ and let $V$ be a compact\nneighborhood of the origin in $G$ such that\n$|\\theta(z)-\\theta(z')|<\\varepsilon$ if $z-z'\\in V$. There are\nfunctions $\\alpha_k\\in\\cc_{\\mathrm{c}}(G)$ with $0\\leq\\alpha_k\\leq1$ such that\n$\\sum_k\\alpha_k=1$ on the support of $\\varphi$ and\n$\\mbox{\\rm supp\\! }\\alpha_k\\subset z_k+V$ for some points $z_k$.  Below we shall\nprove:\n\\begin{equation}\\label{eq:byza}\n\\|T_{XY}(\\varphi)\\theta(Q_Y)- \n{\\textstyle\\sum_k}\\theta(Q_X-z_k)T_{XY}(\\varphi\\alpha_k)\\| \n\\leq\n\\varepsilon\\|T_{XY}(|\\varphi|)\\|.\n\\end{equation}\nThis implies $\\mathscr{T}_{XY}\\cdot\\mathcal{A}|_Y\\subset\\mathcal{A}|_X\\cdot\\mathscr{T}_{XY}$. If we\ntake adjoints, use \\eqref{eq:rad} and interchange $X$ and $Y$ in the\nfinal relation, we obtain $\\mathcal{A}|_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{A}|_Y$\nhence the proposition is proved.  For $u\\in\\cc_{\\mathrm{c}}(X)$ we have:\n\\begin{align*}\n(T_{XY}(\\varphi)\\theta(Q_Y)u)(x) &=\n\\int_Y\\varphi(x-y)\\theta(y)u(y)\\text{d} y \n=\\sum_k\\int_Y\\varphi(x-y)\\alpha_k(x-y)\\theta(y)u(y)\\text{d} y \\\\\n&=\n\\sum_k\\int_Y\\varphi(x-y)\\alpha_k(x-y)\\theta(x-z_k)u(y)\\text{d} y\n+(Ru)(x)\\\\ \n&= \n\\sum_k\\left(\\theta(Q_X-z_k)T_{XY}(\\varphi\\alpha_k)u\\right)(x) \n+(Ru)(x).\n\\end{align*}\nWe can estimate the remainder as follows\n$$\n|(Ru)(x)|=\\left|\\sum_k\\int_Y\\varphi(x-y)\\alpha_k(x-y)\n[\\theta(y)-\\theta(x-z_k)]u(y)\\text{d} y \\right|\\leq\n\\varepsilon\\int_Y|\\varphi(x-y)u(y)|\\text{d} y.\n$$ \nbecause $x-z_k-y\\in V$.  This proves \\eqref{eq:byza}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\n\\begin{proposition}\\label{pr:ryz}\n $\\mathscr{T}_{XY}$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ and\n\\begin{equation}\\label{eq:hyz}\n\\mathscr{T}_{XY}^*\\cdot\\mathscr{T}_{XY}=\\mathscr{C}_Y(X\\cap Y), \\hspace{2mm}\n\\mathscr{T}_{XY}\\cdot\\mathscr{T}_{XY}^*=\\mathscr{C}_X(X\\cap Y).\n\\end{equation}\nThus $\\mathscr{T}_{XY}$ is a $(\\mathscr{C}_X(X\\cap Y),\\mathscr{C}_Y(X\\cap Y))$ imprimitivity\nbimodule.\n\\end{proposition}\n\\proof Due to \\eqref{eq:rad}, to prove the first relation in\n\\eqref{eq:hyz} we have to compute the clspan $\\mathscr{C}$ of the operators\n$T_{XY}(\\varphi)T_{YX}(\\psi)$ with $\\varphi,\\psi$ in $\\cc_{\\mathrm{c}}(G)$.  We\nrecall the notation $G^2=G\\oplus G$, this is a locally compact\nabelian group and $X^2=X\\oplus X$ is a closed subgroup. Let us\nchoose functions $\\varphi_k,\\psi_k\\in\\cc_{\\mathrm{c}}(G)$ and let\n$\\Phi=\\sum_k\\varphi_k\\otimes\\psi_k\\in\\cc_{\\mathrm{c}}(G^2)$. If\n$\\psi_k^\\dag(x)=\\psi_k(-x)$, then $\\sum_k\nT_{XY}(\\varphi_k)T_{YX}(\\psi_k^\\dag)$ is an integral operator on\n$L^2(X)$ with kernel $\\theta_X=\\theta|_{X^2}$ where $\\theta:G^2\\to\n\\mathbb{C}$ is given by\n$$ \n\\theta(x,x')= \\int_Y\\Phi(x+y,x'+y)\\text{d} y.\n$$ Since the set of decomposable functions is dense in $\\cc_{\\mathrm{c}}(G^2)$ in\nthe inductive limit topology, an easy approximation argument shows\nthat $\\mathscr{C}$ contains all integral operators with kernels of the same\nform as $\\theta_X$ but with arbitrary $\\Phi\\in\\cc_{\\mathrm{c}}(G^2)$.  Let\n$Y^{(2)}$ be the closed subgroup of $G^2$ consisting\nof the elements $(y,y)$ with $y\\in Y$. Then $K=\\mbox{\\rm supp\\! }\\Phi\\subset G^2$\nis a compact, $\\theta$ is zero outside $K+Y^{(2)}$, and\n$\\theta(a+b)=\\theta(a)$ for all $a\\in G^2,b\\in Y^{(2)}$. Thus\n$\\theta\\in\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})$, with the usual identification\n$\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(G^2)$. From Proposition 2.48 in\n\\cite{Fo} it follows that reciprocally, any function $\\theta$ in\n$\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})$ can be represented in terms of some $\\Phi$ in\n$\\cc_{\\mathrm{c}}(G^2)$ as above. Thus $\\mathscr{C}$ is the closure of the set of\nintegral operators on $L^2(X)$ with kernels of the form $\\theta_X$\nwith $\\theta\\in\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})$. According to Lemma\n\\ref{lm:double}, the pair of subgroups $X^2,Y^{(2)}$ is regular, so\nwe may apply Lemma \\ref{lm:reg} to get\n$\\cc_{\\mathrm{c}}(G^2\/Y^{(2)})|_{X^2}=\\cc_{\\mathrm{c}}(X^2\/D)$ where $D=X^2\\cap\nY^{(2)}=\\{(x,x)\\mid x\\in X\\cap Y\\}$.  But by Lemma \\ref{lm:sym} the\nnorm closure in $\\mathscr{L}_X$ of the set of integral operators with\nkernel in $\\cc_{\\mathrm{c}}(X^2\/D)$ is $\\mathscr{C}_X\/(X\\cap Y)$. This proves\n\\eqref{eq:hyz}.\n\nIt remains to prove that $\\mathscr{T}_{XY}$ is a Hilbert $C^*$-submodule of\n$\\mathscr{L}_{XY}$, i.e. that we have\n\\begin{equation}\\label{eq:hyz1}\n\\mathscr{T}_{XY}\\cdot\\mathscr{T}_{XY}^*\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}.\n\\end{equation}\nThe first identity in \\eqref{eq:hyz} and \\eqref{eq:cyzc} imply\n\\begin{equation*}\n\\mathscr{T}_{XY}\\cdot\\mathscr{T}_{XY}^*\\cdot\\mathscr{T}_{XY} = \n\\mathscr{T}_{XY}\\cdot \\mathscr{T}_Y\\cdot\\mathcal{C}_Y(X\\cap Y)=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y).\n\\end{equation*}\nFrom Lemma \\ref{lm:reg} we get \n$$\n\\mathcal{C}_Y(X\\cap Y)=\\mathcal{C}_G(X\\cap Y)|_Y =\n\\mathcal{C}_G(X)|_Y \\cdot \\mathcal{C}_G(Y)|_Y = \\mathcal{C}_G(X)|_Y \n$$ because $\\mathcal{C}_G(Y)|_Y=\\mathbb{C}$.  Then by using Proposition\n\\ref{pr:nyza} we obtain\n$$\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y) = \\mathscr{T}_{XY}\\cdot\\mathcal{C}_G(X)|_Y =\n\\mathcal{C}_G(X)|_X \\cdot\\mathscr{T}_{XY} =\\mathscr{T}_{XY}\n$$\nbecause $\\mathcal{C}_G(X)|_X=\\mathbb{C}$. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{corollary}\\label{co:txy}\nWe have\n\\begin{align}\n\\mathscr{T}_{XY} &= \\mathscr{T}_{XY}\\mathscr{T}_Y=\\mathscr{T}_{XY}\\mathcal{C}_Y(X\\cap Y) \n\\label{eq:txy1}\n\\\\\n&= \\mathscr{T}_X\\mathscr{T}_{XY}=\\mathcal{C}_X(X\\cap Y)\\mathscr{T}_{XY}.\n\\label{eq:txy2}\n\\end{align}\n\\end{corollary}\n\\proof If $\\mathscr{M}$ is a Hilbert $\\mathscr{A}$-module then $\\mathscr{M}=\\mathscr{M}\\mathscr{A}$ hence\n$\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\mathscr{C}_Y(X\\cap Y)$ by Proposition \\ref{pr:ryz}. The\nspace $\\mathscr{C}_Y(X\\cap Y)$ is a $\\mathscr{T}_Y$-bimodule and $\\mathscr{C}_Y(X\\cap\nY)=\\mathscr{C}_Y(X\\cap Y)\\cdot \\mathscr{T}_Y$ by \\eqref{eq:Cxy} hence we get\n$\\mathscr{C}_Y(X\\cap Y)=\\mathscr{C}_Y(X\\cap Y)\\mathscr{T}_Y$ by the Cohen-Hewitt theorem.\nThis proves the first equality in \\eqref{eq:txy1} and the other ones\nare proved similarly.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nIf $\\mathcal{G}$ is a set of closed subgroups of $G$ then the\n\\emph{semilattice generated by $\\mathcal{G}$} is the set of finite\nintersections of elements of $\\mathcal{G}$.\n\n\\begin{proposition}\\label{pr:product}\nLet $X,Y,Z$ be closed subgroups of $G$ such that any two subgroups\nfrom the semilattice generated by the family $\\{X,Y,Z\\}$ are\ncompatible. Then:\n\\begin{align}\\label{eq:product}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY} &=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap Z)= \\mathcal{C}_X(X\\cap Z)\\cdot\\mathscr{T}_{XY} \\\\\n&= \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)= \n\\mathcal{C}_X(X\\cap Y\\cap Z)\\cdot\\mathscr{T}_{XY}.\n\\end{align} \nIn particular, if $Z\\supset X\\cap Y$ then\n\\begin{equation}\\label{eq:factor}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}.\n\\end{equation}\n\\end{proposition}\n\\proof We first prove \\eqref{eq:factor} in the particular case\n$Z=G$. As in the proof of Proposition \\ref{pr:ryz} we see that\n$\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{G Y}$ is the the closure in $\\mathscr{L}_{XY}$ of the\nset of integral operators with kernels \n$\\theta_{XY}=\\theta|_{X\\times Y}$ where $\\theta:G^2\\to \\mathbb{C}$ is\ngiven by \n$$ \n\\theta(x,y)= \\int_G\\sum_k\\varphi_k(x-z)\\psi_k(z-y)\\text{d} z=\n\\int_G\\sum_k\\varphi_k(x-y-z)\\psi_k(z)\\text{d} z\\equiv\\xi(x-y)\n$$ where $\\varphi_k,\\psi_k\\in\\cc_{\\mathrm{c}}(G)$ and $\\xi=\\sum_k\\varphi_k*\\psi_k$\nconvolution product on $G$. Since $\\cc_{\\mathrm{c}}(G)*\\cc_{\\mathrm{c}}(G)$ is dense in\n$\\cc_{\\mathrm{c}}(G)$ in the inductive limit topology, the space\n$\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{G Y}$ is the the closure of the set of integral\noperators with kernels $\\theta(x,y)=\\xi(x-y)$ with $\\xi\\in\\cc_{\\mathrm{c}}(G)$.\nBy Proposition \\ref{pr:def2} this is  $\\mathscr{T}_{XY}$. \n\nNow we prove \\eqref{eq:product}. From \\eqref{eq:factor} with $Z=G$\nand \\eqref{eq:hyz} we get:\n\\begin{equation*}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY} =\n\\mathscr{T}_{XG}\\cdot\\mathscr{T}_{G Z}\\cdot\\mathscr{T}_{ZG}\\cdot\\mathscr{T}_{G Y}\n= \\mathscr{T}_{XG}\\cdot\\mathcal{C}_G(Z)\\cdot \\mathscr{T}_G\\cdot\\mathscr{T}_{GY}.\n\\end{equation*}\nThen from Proposition \\eqref{pr:nyza} and Lemma \\ref{lm:reg} we get:\n$$\n\\mathcal{C}_G(Z)\\cdot \\mathscr{T}_G\\cdot\\mathscr{T}_{GY}=\\mathcal{C}_G(Z)\\cdot\\mathscr{T}_{G Y}=\n\\mathscr{T}_{G Y}\\cdot\\mathcal{C}_G(Z)|_Y= \\mathscr{T}_{G Y}\\cdot\\mathcal{C}_Y(Y\\cap Z).\n$$ \nWe obtain \\eqref{eq:product} by using once again\n\\eqref{eq:factor} with $Z=G$ and taking adjoints. On the other hand,\nthe relation $\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y)$ holds because\nof \\eqref{eq:txy1}, so we have\n$$\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap Z)=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y)\\cdot\\mathcal{C}_Y(Y\\cap Z)=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\n$$ where we also used \\eqref{eq:reg1} and the fact that $X\\cap Y$,\n$Z\\cap Y$ are compatible. Finally, to get \\eqref{eq:factor} for\n$Z\\supset X\\cap Y$ we use once again \\eqref{eq:hyz}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nThe object of main interest for us is introduced in the next\ndefinition. \n\n\\smallskip\n\n\\begin{definition}\\label{df:nxyz}\nIf $X,Y$ are compatible subgroups and $Z$ is a closed subgroup of\n$X\\cap Y$ then we set\n\\begin{equation}\\label{eq:nxyz}\n\\mathscr{C}_{XY}(Z):=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Z)=\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}.\n\\end{equation} \n\\end{definition}\nThe equality above follows from \\eqref{eq:ayza} with $\\mathcal{A}=\\mathcal{C}_G(Z)$.\nWe clearly have $\\mathscr{C}_{XY}(X\\cap Y)=\\mathscr{T}_{XY}$ and\n$\\mathscr{C}_{XX}(Y)=\\mathscr{C}_X(Y)$ if $X\\supset Y$. Moreover\n\\begin{equation}\\label{eq:nadj}\n\\mathscr{C}_{XY}^*(Z):=\\mathscr{C}_{XY}(Z)^*=\\mathscr{C}_{YX}(Z)\n\\end{equation}\nbecause of \\eqref{eq:rad}.\n\n\n\\begin{theorem}\\label{th:nxyz}\n$\\mathscr{C}_{XY}(Z)$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ such that\n\\begin{equation}\\label{eq:nnz}\n\\mathscr{C}_{XY}^*(Z)\\cdot\\mathscr{C}_{XY}(Z)=\\mathscr{C}_Y(Z)\n\\hspace{2mm}\\text{and}\\hspace{2mm}\n\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{XY}^*(Z)=\\mathscr{C}_X(Z).\n\\end{equation} \nIn particular, $\\mathscr{C}_{XY}(Z)$ is a\n$(\\mathscr{C}_X(Z),\\mathscr{C}_Y(Z))$ imprimitivity bimodule.\n\\end{theorem}\n\\proof \nBy using \\eqref{eq:nadj}, the definition \\eqref{eq:nxyz}, and\n\\eqref{eq:reg1} we get\n\\begin{align*}\n\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{YX}(Z) &=\n\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}\\cdot \\mathscr{T}_{YX}\\cdot\\mathcal{C}_X(Z)\\\\\n&=\n\\mathcal{C}_X(Z)\\cdot\\mathcal{C}_X(X\\cap Y)\\cdot \\mathscr{T}_X\\cdot\\mathcal{C}_X(Z)\\\\\n&=\n\\mathcal{C}_X(Z)\\cdot \\mathscr{T}_X\\cdot\\mathcal{C}_X(Z)= \\mathcal{C}_X(Z)\\cdot \\mathscr{T}_X\n\\end{align*}\nwhich proves the second equality in \\eqref{eq:nnz}.  The first one\nfollows by interchanging $X$ and $Y$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\n\n\\subsection{Many-body systems}\n\\label{ss:gaz} \n\nHere we give a formal definition of the notion of ``many-body\nsystem'' then define and discuss the Hamiltonian algebra associated\nto it.\n\n\nLet $\\mathscr{S}$ be a set of locally compact abelian groups with the\nfollowing property: for any $X,Y\\in\\mathscr{S}$ there is $Z\\in\\mathscr{S}$ such that\n$X$ and $Y$ are compatible subgroups of $Z$. Note that this implies\nthe following: if $Y\\subset X$ then the topology and the group\nstructure of $Y$ coincide with those induced by $X$.\n\nIf $\\mathscr{S}$ is a set of $\\sigma$-compact locally compact abelian groups\nthen the compatibility assumption is equivalent to the following\nmore explicit condition: for any $X,Y\\in\\mathscr{S}$ there is $Z\\in\\mathscr{S}$ such\nthat $X$ and $Y$ are closed subgroups of $Z$ and $X+Y$ is closed in\n$Z$. \n\n\n\\begin{definition}\\label{df:mb}\nA \\emph{many-body system} is a couple $(\\mathcal{S},\\lambda)$ where:\n\\begin{compactenum}\n\\item[(i)] $\\mathcal{S}\\subset\\mathscr{S}$ is a subset such that \n$X,Y\\in\\mathcal{S}\\Rightarrow X\\cap Y\\in\\mathcal{S}$ and if\n$X\\supsetneq Y$ then $X\/Y$ is not compact,\n\\item[(ii)] \n$\\lambda$ is a map $X\\mapsto\\lambda_X$ which associates a Haar\nmeasures $\\lambda_X$ on $X$ to each $X\\in\\mathcal{S}$. \n\\end{compactenum}\n\\end{definition}\n\nWe identify $\\mathcal{S}=(\\mathcal{S},\\lambda)$ so the choice of Haar measures is\nimplicit. Note that the Hilbert space $\\mathcal{H}_\\mathcal{S}$ and the\n$C^*$-algebra $\\mathscr{C}_\\mathcal{S}$ that we introduce below depend on $\\lambda$\nbut different choices give isomorphic objects.  Each $X\\in\\mathcal{S}$ is\nequipped with a Haar measure so the Hilbert spaces $\\mathcal{H}_X= L^2(X)$\nare well defined. If $Y\\subset X$ are in $\\mathcal{S}$ then $X\/Y$ is\nequipped with the quotient measure so $\\mathcal{H}_{X\/Y}= L^2(X\/Y)$ is well\ndefined.\n\n\n{\\bf Example:} Let $\\mathscr{S}$ the set of all finite dimensional vector\nsubspaces of a vector space over an infinite locally compact field\nand let $\\mathcal{S}$ be any subset of $\\mathscr{S}$ such that $X,Y\\in\\mathcal{S}\\Rightarrow\nX\\cap Y \\in \\mathcal{S}$. \n\n\nFor each $X\\in\\mathcal{S}$ let $\\mathcal{S}_X$ be the set of $Y\\in\\mathcal{S}$ such that\n$Y\\subset X$.  This is an $N$-body system with $X$ as configuration\nspace in the sense of Definition \\ref{df:nb}.  Then by Lemma\n\\ref{lm:reg} the space\n\\begin{equation}\\label{eq:sax}\n\\mathcal{C}_X := {\\textstyle\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}_X}}\\mathcal{C}_X(Y)\n\\end{equation}\nis an $X$-algebra so the crossed product $\\mathcal{C}_X\\rtimes X$ is well\ndefined and we clearly have\n\\begin{equation}\\label{eq:crsax}\n\\mathscr{C}_X := \\mathcal{C}_X\\rtimes X \\equiv \\mathcal{C}_X\\cdot\\mathscr{T}_X =\n{\\textstyle\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}_X}}\\mathscr{C}_X(Y).\n\\end{equation}\nThe $C^*$-algebra $\\mathscr{C}_X$ is realized on the Hilbert space $\\mathcal{H}_X$\nand we think of it as the Hamiltonian algebra of the $N$-body system\ndetermined by $\\mathcal{S}_X$.\n\n\\begin{theorem}\\label{th:grsax}\nThe $C^*$-algebras $\\mathcal{C}_X$ and $\\mathscr{C}_X$ are $\\mathcal{S}_X$-graded by \nthe decompositions \\eqref{eq:sax} and \\eqref{eq:crsax}. \n\\end{theorem}\n\nThe theorem is a particular case of results due to A. Mageira,\ncf. Propositions 6.1.2, 6.1.3 and 4.2.1 in \\cite{Ma} (or see\n\\cite{Ma3}). We mention that the results in \\cite{Ma,Ma3} are much\ndeeper since the groups are allowed to be noncommutative and the\ntreatment is so that the second part of condition (i) is not\nneeded. The case when $\\mathcal{S}$ consists of linear subspaces of a finite\ndimensional real vector space has been considered in \\cite{BG1,DG1}\nand the corresponding version of Theorem \\ref{th:grsax} is proved\nthere by elementary means.\n\n\n\\begin{definition}\\label{df:main}\nIf $X,Y\\in\\mathcal{S}$ then\n$\\mathscr{C}_{XY} := \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y=\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}$.\n\\end{definition}\n\nIn particular $\\mathscr{C}_{XX}=\\mathscr{C}_X$ is a $C^*$-algebra of operators on\n$\\mathcal{H}_X$.  For $X\\neq Y$ the space $\\mathscr{C}_{XY}$ is a closed linear\nspace of operators $\\mathcal{H}_Y\\to \\mathcal{H}_X$ canonically associated to the\nsemilattice of groups $\\mathcal{S}_{X\\cap Y}$, cf. \\eqref{eq:main}. We call\nthese spaces \\emph{coupling modules} because they are Hilbert\n$C^*$-modules and determine the way the systems corresponding to $X$\nand $Y$ are allowed to interact.\n\n\n\nFor each pair $X,Y\\in\\mathcal{S}$ with $X\\supset Y$ we set \n\\begin{equation}\\label{eq:saX}\n\\mathcal{C}^Y_X :=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_Y}}\\mathcal{C}_X(Z).\n\\end{equation}\nThis is also an $X$-algebra so we may define $\\mathscr{C}^Y_X=\\mathcal{C}^Y_X\\rtimes\nX$ and we have\n\\begin{equation}\\label{eq:saX1}\n\\mathscr{C}^Y_X := \\mathcal{C}^Y_X\\rtimes X=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_Y}}\\mathscr{C}_X(Z).\n\\end{equation}\nIf $X=Y\\oplus Z$ then $\\mathcal{C}_X^Y\\simeq\\mathcal{C}_Y\\otimes 1$ and \n$\\mathscr{C}_X^Y\\simeq\\mathscr{C}_Y\\otimes \\mathscr{T}_Z$. \n\n\\begin{lemma}\\label{lm:xyprod}\nLet $X\\in\\mathcal{S}$ and $Y\\in\\mathcal{S}_X$. Then\n\\begin{equation}\\label{eq:xy1}\n\\mathcal{C}_X^{Y}=\\mathcal{C}_X(Y)\\cdot\\mathcal{C}_X \\hspace{2mm}\\text{and}\\hspace{2mm}\n\\mathscr{C}_X^{Y}=\\mathcal{C}_X(Y)\\cdot\\mathscr{C}_X=\\mathscr{C}_X\\cdot\\mathcal{C}_X(Y).\n\\end{equation}\nMoreover, for all  $Y,Z\\in\\mathcal{S}_X$ we have\n\\begin{equation}\\label{eq:xy2}\n\\mathcal{C}_X^Y\\cdot\\mathcal{C}_X^Z=\\mathcal{C}_X^{Y\\cap Z} \n\\hspace{2mm}\\text{and}\\hspace{2mm} \n\\mathscr{C}_X^Y\\cdot\\mathscr{C}_X^Z=\\mathscr{C}_X^{Y\\cap Z}.\n\\end{equation}\n\\end{lemma}\n\\proof The abelian case follows from \\eqref{eq:reg1} and a\nstraightforward computation. For the crossed product algebras we use\n$\\mathcal{C}_X(Y)\\cdot\\mathscr{C}_X=\\mathcal{C}_X(Y)\\cdot\\mathcal{C}_X\\cdot \\mathscr{T}_X$ and the first\nrelation in \\eqref{eq:xy1} for example.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{lemma}\\label{lm:nxy}\nFor arbitrary $X,Y\\in\\mathcal{S}$ we have\n\\begin{equation}\\label{eq:main}\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y\n=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}=\n\\mathcal{C}_X^{X\\cap Y}\\cdot\\mathscr{T}_{XY}.\n\\end{equation}\n\\end{lemma}\n\\proof\nIf $G\\in\\mathscr{S}$ contains $X\\cup Y$ then clearly\n$$\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_X} }\\mathcal{C}_X(Z)\\cdot\\mathscr{T}_{XY}=\n{\\textstyle\\sum^\\mathrm{c}_{Z\\in\\mathcal{S}_X} }\\mathcal{C}_G(Z)|_X\\cdot\\mathscr{T}_{XY}.\n$$\nFrom \\eqref{eq:ayza} and \\eqref{eq:reg2} we get\n$$\n\\mathcal{C}_G(Z)|_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap Z).\n$$ \nSince $Y\\cap Z$ runs over $\\mathcal{S}_{X\\cap Y}$ when $Z$ runs over\n$\\mathcal{S}_X$ we obtain $\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}$.\nSimilarly $\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y=\\mathcal{C}_X^{X\\cap Y}\\cdot\\mathscr{T}_{XY}$. \nOn the other hand $\\mathcal{C}_X^{X\\cap Y}=\\mathcal{C}_G^{X\\cap Y}|_X$ and similarly\nwith $X,Y$ interchanged, hence \n$\\mathcal{C}_X^{X\\cap Y}\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}$\nbecause of \\eqref{eq:ayza}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\\begin{proposition}\\label{pr:mxyz}\nLet $X,Y,Z\\in\\mathcal{S}$. Then $\\mathscr{C}_{XY}^*=\\mathscr{C}_{YX}$ and\n\\begin{equation}\\label{eq:mxyz}\n\\mathscr{C}_{XZ}\\cdot\\mathscr{C}_{ZY}=\\mathscr{C}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y\\cap Z}=\n\\mathcal{C}_X^{X\\cap Y\\cap Z}\\cdot\\mathscr{C}_{XY} \\subset \\mathscr{C}_{XY}. \n\\end{equation}\nIn particular $\\mathscr{C}_{XZ}\\cdot\\mathscr{C}_{ZY}=\\mathscr{C}_{XY}$ if $Z\\supset X\\cap Y$.\n\\end{proposition}\n\\proof \nThe first assertion follows from \\eqref{eq:rad}.  From the\nDefinition \\ref{df:main} and Proposition \\ref{pr:product} we then get\n\\begin{align*}\n\\mathscr{C}_{XZ}\\cdot\\mathscr{C}_{ZY} &=\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}\\cdot\\mathcal{C}_Y =\n\\mathcal{C}_X\\cdot\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y \\\\ &=\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y =\n\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y.\n\\end{align*}\nBut $\\mathcal{C}_Y(X\\cap Y\\cap Z)\\cdot\\mathcal{C}_Y=\\mathcal{C}_Y^{X\\cap Y\\cap Z}$ by Lemma\n\\ref{lm:xyprod}. For the last inclusion in \\eqref{eq:mxyz} we use\nthe obvious relation $\\mathcal{C}_Y^{X\\cap Y\\cap Z}\\cdot\\mathcal{C}_Y\\subset\\mathcal{C}_Y$.\nThe last assertion of the proposition follows from \\eqref{eq:main}. \n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nThe following theorem is a consequence of the results obtained so\nfar.\n\n\\begin{theorem}\\label{th:nmod}\n$\\mathscr{C}_{XY}$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ such that \n\\begin{equation}\\label{eq:nmod}\n\\mathscr{C}_{XY}^*\\cdot\\mathscr{C}_{XY}=\\mathscr{C}_Y^{X\\cap Y} \\text{ and }\n\\mathscr{C}_{XY}\\cdot\\mathscr{C}_{XY}^*=\\mathscr{C}_X^{X\\cap Y}.\n\\end{equation}\nIn particular, $\\mathscr{C}_{XY}$ is a\n$(\\mathscr{C}_X^{X\\cap Y},\\mathscr{C}_Y^{X\\cap Y})$ imprimitivity bimodule. \n\\end{theorem}\n\nWe recall the conventions\n\\begin{align}\n& X,Y\\in\\mathcal{S} \\text{ and } Y\\not\\subset X \\Rightarrow \n\\mathcal{C}_X(Y)= \\mathscr{C}_X(Y)=\\{0\\}, \\label{eq:convn} \\\\\n& X,Y,Z\\in\\mathcal{S} \\text{ and } Z\\not\\subset X\\cap Y \\Rightarrow\n\\mathscr{C}_{XY}(Z)=\\{0\\}.\\label{eq:convn1}\n\\end{align}\nFrom now on by ``graded'' we mean $\\mathcal{S}$-graded.  Then\n$\\mathscr{C}_X=\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}}\\mathscr{C}_X(Y)$ is a graded\n$C^*$-algebras supported by the ideal $\\mathcal{S}_X$ of $\\mathcal{S}$, in\nparticular it is a graded ideal in $\\mathscr{C}_X$.  With the notations of\nSubsection \\ref{ss:grca} the algebra $\\mathscr{C}^Y_X=\\mathscr{C}_X(\\mathcal{S}_Y)$ is a\ngraded ideal of $\\mathscr{C}_X$ supported by $\\mathcal{S}_Y$. Similarly for $\\mathcal{C}_X$\nand $\\mathcal{C}_X^Y$.\n\nSince $\\mathscr{C}_X^{X\\cap Y}$ and $\\mathscr{C}_Y^{X\\cap Y}$ are ideals in $\\mathscr{C}_X$\nand $\\mathscr{C}_Y$ respectively, Theorem \\ref{th:nmod} allows us to equip\n$\\mathscr{C}_{XY}$ with (right) Hilbert $\\mathscr{C}_Y$-module and left Hilbert\n$\\mathscr{C}_X$-module structures (which are not full in general).\n\n\\begin{theorem}\\label{th:nmain}\nThe Hilbert $\\mathscr{C}_Y$-module $\\mathscr{C}_{XY}$ is graded by the family of\n$C^*$-submodules  $\\{\\mathscr{C}_{XY}(Z)\\}_{Z\\in\\mathcal{S}}$.\n\\end{theorem}\n\\proof We use Proposition \\ref{pr:rhm} with $\\mathscr{M}=\\mathscr{T}_{XY}$ and\n$\\mathcal{C}_Y(Z)$ as algebras $\\mathcal{C}(\\sigma)$. Then $\\mathscr{A}=\\mathscr{C}_Y(X\\cap Y)$ by\n\\eqref{eq:hyz} hence $\\mathscr{A}\\cdot\\mathcal{C}_Y(Z)=\\mathscr{C}_Y(Z)$ and the conditions\nof the proposition are satisfied.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\\begin{remark}\\label{re:precise}\nThe following more precise statement is a consequence of the Theorem\n\\ref{th:nmain}: the Hilbert $\\mathscr{C}_Y^{X\\cap Y}$-module $\\mathscr{C}_{XY}$ is\n$\\mathcal{S}_{X\\cap Y}$-graded by the family of $C^*$-submodules\n$\\{\\mathscr{C}_{XY}(Z)\\}_{Z\\in\\mathcal{S}_{X\\cap Y}}$.\n\\end{remark}\n\nFinally, we may construct the $C^*$-algebra $\\mathscr{C}$ which is of main\ninterest for us, the many-body Hamiltonian algebra. We shall\ndescribe it as an algebra of operators on the Hilbert space\n\\begin{equation}\\label{eq:bigh}\n\\mathcal{H}\\equiv\\mathcal{H}_\\mathcal{S}={\\textstyle\\oplus_{X\\in\\mathcal{S}}} \\mathcal{H}_X\n\\end{equation}\nwhich is a kind of Boltzmann-Fock space (without symmetrization or\nanti-symmetrization) determined by the semilattice $\\mathcal{S}$. Note that\nif the zero group $O=\\{0\\}$ belongs to $\\mathcal{S}$ then $\\mathcal{H}$ contains\n$\\mathcal{H}_O=\\mathbb{C}$ as a subspace, this is the vacuum sector.  Let\n$\\Pi_{X}$ be the orthogonal projection of $\\mathcal{H}$ onto $\\mathcal{H}_X$ and let\nus think of its adjoint $\\Pi_{X}^*$ as the natural embedding\n$\\mathcal{H}_X\\subset\\mathcal{H}$. Then for any pair $X,Y\\in\\mathcal{S}$ we identify\n\\begin{equation}\\label{eq:identc}\n\\mathscr{C}_{XY}\\equiv\\Pi^*_{X}\\mathscr{C}_{XY}\\Pi_{Y} \\subset L(\\mathcal{H}).\n\\end{equation}\nThus we realize $\\{\\mathscr{C}_{XY}\\}_{X,Y\\in\\mathcal{S}}$ as a linearly independent\nfamily of closed subspaces of $L(\\mathcal{H})$ such that\n$\\mathscr{C}_{XY}^*=\\mathscr{C}_{YX}$ and $\\mathscr{C}_{XZ}\\mathscr{C}_{Z'Y}\\subset\\mathscr{C}_{XY}$ for all\n$X,Y,Z,Z'\\in\\mathcal{S}$. Then by what we proved before, especially\nProposition \\ref{pr:mxyz}, the space\n$\\sum\\nolimits_{X,Y\\in\\mathcal{S}}\\mathscr{C}_{XY}$ is a $*$-subalgebra of $L(\\mathcal{H})$\nhence its closure\n\\begin{equation}\\label{eq:bigco}\n\\mathscr{C}\\equiv\\mathscr{C}_\\mathcal{S}= {\\textstyle\\sum^\\mathrm{c}_{X,Y\\in\\mathcal{S}}}\\mathscr{C}_{XY}.\n\\end{equation}\nis a $C^*$-algebra of operators on $\\mathcal{H}$.  Note that one may view\n$\\mathscr{C}$ as a matrix $(\\mathscr{C}_{XY})_{X,Y\\in\\mathcal{S}}$.  \n\nIn a similar way one may associate to the spaces $\\mathscr{T}_{XY}$ a\nclosed self-adjoint subspace $\\mathscr{T}\\subset L(\\mathcal{H})$.  It is also useful\nto define a new subspace $\\mathscr{T}^\\circ\\subset L(\\mathcal{H})$ by\n$\\mathscr{T}^\\circ_{XY}=\\mathscr{T}_{XY}$ if $X\\sim Y$ and $\\mathscr{T}^\\circ=\\{0\\}$ if\n$X\\not\\sim Y$. Here $X\\sim Y$ means $X\\subset Y$ or $Y\\subset X$\n. Clearly $\\mathscr{T}^\\circ$ is a closed self-adjoint linear subspace of\n$\\mathscr{T}$.  Finally, let $\\mathcal{C}$ be the diagonal $C^*$-algebra\n$\\mathcal{C}\\equiv\\oplus_X\\mathcal{C}_X$ of operators on $\\mathcal{H}$.\n\n\\begin{theorem}\\label{th:tc}\nWe have $\\mathscr{C}=\\mathscr{T}\\cdot\\mathcal{C}=\\mathcal{C}\\cdot\\mathscr{T}=\\mathscr{T}\\cdot\\mathscr{T}=\n\\mathscr{T}^\\circ\\cdot\\mathscr{T}^\\circ$.\n\\end{theorem}\n\\proof The first two equalities are an immediate consequence of the\nDefinition \\ref{df:main}. To prove the third equality we use\nProposition \\ref{pr:product}, more precisely the relation\n\\[\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(X\\cap Y\\cap Z)=\n\\mathscr{C}_{XY}(X\\cap Y\\cap Z)\n\\]\nwhich holds for any $X,Y,Z$. Then\n\\[\n{\\textstyle\\sum^\\mathrm{c}_Z}\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\n{\\textstyle\\sum^\\mathrm{c}_Z}\\mathscr{C}_{XY}(X\\cap Y\\cap Z)=\n{\\textstyle\\sum^\\mathrm{c}_Z}\\mathscr{C}_{XY}(Z)=\\mathscr{C}_{XY}\n\\]\nwhich is equivalent to $\\mathscr{T}\\cdot\\mathscr{T}=\\mathscr{C}$. Now we prove the last\nequality in the proposition. We have\n\\[\n{\\textstyle\\sum^\\mathrm{c}_Z} \\mathscr{T}^\\circ_{XZ}\\cdot\\mathscr{T}^\\circ_{ZY}= \n\\text{ closure of the sum }\n{\\textstyle\\sum_{\\substack{Z\\sim X\\\\ Z\\sim Y}}}\n\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}. \n\\]\nIn the last sum we have four possibilities: $Z\\supset X\\cup Y$,\n$X\\supset Z\\supset Y$, $Y\\supset Z\\supset X$, and $Z\\subset X\\cap\nY$. In the first three cases we have $Z\\supset X\\cap Y$ hence\n$\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}$ by \\eqref{eq:factor}. In the last\ncase we have $\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Z)$ by\n\\eqref{eq:product}. This proves $\\mathscr{T}^\\circ\\cdot\\mathscr{T}^\\circ=\\mathscr{C}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nFinally, we are able to equip $\\mathscr{C}$ with an $\\mathcal{S}$-graded\n$C^*$-algebra structure.\n\n\\begin{theorem}\\label{th:cgrad}\nFor each $Z\\in\\mathcal{S}$ the space $\\mathscr{C}(Z):=\n\\sum^\\mathrm{c}_{X,Y\\in\\mathcal{S}}\\mathscr{C}_{XY}(Z)$ is a $C^*$-subalgebra of\n$\\mathscr{C}$. The family $\\{\\mathscr{C}(Z)\\}_{Z\\in\\mathcal{S}}$ defines a graded\n$C^*$-algebra structure on $\\mathscr{C}$.\n\\end{theorem}\n\\proof \nWe first prove the following relation:\n\\begin{equation}\\label{eq:xyzef}\n\\mathscr{C}_{XZ}(E)\\cdot\\mathscr{C}_{ZY}(F)=\\mathscr{C}_{XY}(E\\cap F)\n\\quad \\text{if } X,Y,Z\\in\\mathcal{S} \\text{ and } E\\subset{X\\cap Z},\nF\\subset{Y\\cap Z}.\n\\end{equation}\nFrom Definition \\ref{df:nxyz}, Proposition \\ref{pr:product},\nrelations \\eqref{eq:reg1} and \\eqref{eq:ayza}, and \n$F\\subset Y\\cap Z$, we get\n\\begin{align*}\n\\mathscr{C}_{XZ}(E)\\cdot\\mathscr{C}_{ZY}(F) &=\n\\mathcal{C}_X(E)\\cdot\\mathscr{T}_{XZ}\\cdot\\mathscr{T}_{ZY}\\cdot\\mathcal{C}_Y(F) \\\\\n&= \\mathcal{C}_X(E)\\cdot\\mathscr{T}_{XY}\\cdot\\mathcal{C}_{Y}(Y\\cap Z)\\cdot\\mathcal{C}_Y(F) \\\\\n&= \\mathcal{C}_X(E)\\cdot\\mathscr{T}_{XY}\\cdot\\mathcal{C}_{Y}(F) \\\\\n&= \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap E)\\cdot\\mathcal{C}_{Y}(F) \\\\\n&= \\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y(Y\\cap E\\cap F).\n\\end{align*}\nAt the next to last step we used $\\mathcal{C}_X(E)=\\mathcal{C}_G(E)|_X$ for some\n$G\\in\\mathscr{S}$ containing both $X$ and $Y$ and then \\eqref{eq:ayza},\n\\eqref{eq:reg2}. Finally, we use $\\mathcal{C}_Y(Y\\cap E\\cap F)=\\mathcal{C}_Y(E\\cap\nF)$ and the Definition \\ref{df:nxyz}. This proves \\eqref{eq:xyzef}.\nDue to the conventions \\eqref{eq:convn}, \\eqref{eq:convn1} we now\nget from \\eqref{eq:xyzef} for $E,F\\in\\mathcal{S}$\n\\[\n{\\textstyle\\sum_{Z\\in\\mathcal{S}}}\\mathscr{C}_{XZ}(E)\\cdot\\mathscr{C}_{ZY}(F)=\n\\mathscr{C}_{XY}(E\\cap F).\n\\]\nThus $\\mathscr{C}(E)\\mathscr{C}(F)\\subset\\mathscr{C}(E\\cap F)$, in particular $\\mathscr{C}(E)$ is a\n$C^*$-algebra. It remains to be shown that the family of\n$C^*$-algebras $\\{\\mathscr{C}(E)\\}_{E\\in\\mathcal{S}}$ is linearly independent. Let\n$A(E)\\in\\mathscr{C}(E)$ such that $A(E)=0$ but for a finite number of $E$\nand assume that $\\sum_E A(E)=0$. Then for all $X,Y\\in\\mathcal{S}$\nwe have $\\sum_E \\Pi_X A(E) \\Pi_Y^* =0$. Clearly \n$\\Pi_X A(E) \\Pi_Y^*\\in\\mathscr{C}_{XY}(E)$ hence from Theorem \\ref{th:nmain}\nwe get $\\Pi_X A(E) \\Pi_Y^*=0$ for all $X,Y$ so $A(E)=0$ for all $E$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\subsection{Subsystems}\n\\label{ss:T} \n\nWe now point out some interesting subalgebras of\n$\\mathscr{C}$. If $\\mathcal{T}\\subset\\mathcal{S}$ is any subset let\n\\begin{equation}\\label{eq:t}\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}\\equiv{\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY} \\quad\n\\text{and} \\quad \\mathcal{H}_\\mathcal{T}\\equiv\\oplus_{X\\in\\mathcal{T}}\\mathcal{H}_X.\n\\end{equation}\nNote that the sum defining $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is already closed if $\\mathcal{T}$ is\nfinite and that $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is a $C^*$-algebra which lives on the\nsubspace $\\mathcal{H}_\\mathcal{T}$ of $\\mathcal{H}$. In fact, if $\\Pi_\\mathcal{T}$ is the orthogonal\nprojection of $\\mathcal{H}$ onto $\\mathcal{H}_\\mathcal{T}$ then\n\\begin{equation}\\label{eq:tt}\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}=\\Pi_\\mathcal{T}\\mathscr{C}_\\mathcal{S}\\Pi_\\mathcal{T}\n\\end{equation}\nand this is a $C^*$-algebra because $\\mathscr{C}\\Pi_\\mathcal{T}\\mathscr{C}\\subset\\mathscr{C}$ by\nProposition \\ref{pr:mxyz}. \nIt is easy to check that $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is a graded $C^*$-subalgebra of\n$\\mathscr{C}$ supported by the ideal $\\textstyle{\\bigcup}_{X\\in\\mathcal{T}}\\mathcal{S}_X$ generated by\n$\\mathcal{T}$ in $\\mathcal{S}$. Indeed, we have\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}\\,\\textstyle{\\bigcap}\\,\\mathscr{C}(E)=\n\\left({\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY}\\right) \\textstyle{\\bigcap}\n\\left({\\textstyle\\sum_{X,Y\\in\\mathcal{S}}^\\mathrm{c}}\\mathscr{C}_{XY}(E)\\right) =\n{\\textstyle\\sum_{X,Y\\in\\mathcal{T}}^\\mathrm{c}}\\mathscr{C}_{XY}(E).\n\\]\nIt is clear that $\\mathscr{C}$ is the inductive limit of the increasing\nfamily of $C^*$-algebras $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ with finite $\\mathcal{T}$.\n\nIf $\\mathcal{T}=\\{X\\}$ then $\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ is just $\\mathscr{C}_X$. If\n$\\mathcal{T}=\\{X,Y\\}$ with distinct $X,Y$ we get a simple but nontrivial\nsituation. Indeed, we shall have $\\mathcal{H}_\\mathcal{T}=\\mathcal{H}_X\\oplus\\mathcal{H}_Y$ and\n$\\mathscr{C}^\\mathcal{T}_\\mathcal{S}$ may be thought as a matrix\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}=\n\\begin{pmatrix}\n\\mathscr{C}_X & \\mathscr{C}_{XY}\\\\\n\\mathscr{C}_{YX} & \\mathscr{C}_Y\n\\end{pmatrix}.\n\\]\nThe grading is now explicitly defined as follows: \n\\begin{compactenum}\n\\item\nIf  $E\\subset X\\cap Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(E)=\n\\begin{pmatrix}\n\\mathscr{C}_X(E) & \\mathscr{C}_{XY}(E)\\\\\n\\mathscr{C}_{YX}(E) & \\mathscr{C}_Y(E)\n\\end{pmatrix}.\n\\]\n\\item   \\label{p:2ex}\nIf  $E\\subset X$ and $E\\not\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(E)=\n\\begin{pmatrix}\n\\mathscr{C}_X(E) & 0\\\\\n0 & 0\n\\end{pmatrix}.\n\\]\n\\item\nIf  $E\\not\\subset X$ and $E\\subset Y$ then\n\\[\n\\mathscr{C}^\\mathcal{T}_\\mathcal{S}(E)=\n\\begin{pmatrix}\n0 & 0\\\\\n0 & \\mathscr{C}_Y(E)\n\\end{pmatrix}.\n\\]\n\\end{compactenum}\n\n\nThe case when $\\mathcal{T}$ is of the form $\\mathcal{S}_X$ for some $X\\in\\mathcal{S}$ is\nespecially interesting.  We denote $\\mathscr{C}_X^\\#\\equiv\\mathscr{C}_{\\mathcal{S}_X}$ and\nwe say that the $\\mathcal{S}_X$-graded $C^*$-algebra is the\n\\emph{unfolding} of the algebra $\\mathscr{C}_X$.  More explicitly\n\\begin{equation}\\label{eq:xc}\n\\mathscr{C}_X^\\#\\equiv{\\textstyle\\sum^\\mathrm{c}_{Y,Z\\in\\mathcal{S}_X}}\\mathscr{C}_{YZ}.\n\\end{equation}\nThe self-adjoint operators affiliated to $\\mathscr{C}_X$ live on the Hilbert\nspace $\\mathcal{H}_X$ and are (an abstract version of) Hamiltonians of an\n$N$-particle system $\\mathscr{S}$ with a fixed $N$ (the configuration space\nis $X$ and $N$ is the number of levels of the semilattice\n$\\mathcal{S}_X$). The unfolding $\\mathscr{C}_X^\\#$ lives on the ``Boltzmann-Fock\nspace'' $\\mathcal{H}_{\\mathcal{S}_X}$ and is obtained by adding interactions which\ncouple the subsystems of $\\mathcal{S}$ which have the groups $Y\\in\\mathcal{S}_X$ as\nconfiguration spaces and $\\mathscr{C}_Y$ as Hamiltonian algebras.\n\nClearly $\\mathscr{C}_X^\\#\\subset\\mathscr{C}_Y^\\#$ if $X\\subset Y$ and $\\mathscr{C}$ is the\ninductive limit of the algebras $\\mathscr{C}_X^\\#$.  Below we give an\ninteresting alternative description of $\\mathscr{C}_X^\\#$.\n\n\\begin{theorem}\\label{th:mor}\nLet $\\mathscr{N}_X=\\oplus_{Y\\in\\mathcal{S}_X}\\mathscr{C}_{YX}$ be the direct sum of the\nHilbert $\\mathscr{C}_X$-modules $\\mathscr{C}_{YX}$ equipped with the direct sum graded\nstructure. Then $\\mathcal{K}(\\mathscr{N}_X) \\cong \\mathscr{C}_X^\\#$ the isomorphism being such\nthat the graded structure on $\\mathcal{K}(\\mathscr{N}_X)$ defined in Theorem\n\\ref{th:kghm} is transported into that of $\\mathscr{C}_X^\\#$.  In other terms,\n$\\mathscr{C}_X^\\#$ is the imprimitivity algebra of the full Hilbert\n$\\mathscr{C}_X$-module $\\mathscr{N}_X$ and $\\mathscr{C}_X$ and $\\mathscr{C}_X^\\#$ are Morita\nequivalent.\n\\end{theorem}\n\\proof If $Y\\subset X$ then $\\mathscr{C}^*_{YX}\\cdot\\mathscr{C}_{YX}=\\mathscr{C}_X^Y$ and\n$\\mathscr{C}_{YX}$ is a full Hilbert $\\mathscr{C}_X^Y$-module. Since the $\\mathscr{C}_X^Y$\nare ideals in $\\mathscr{C}_X$ and their sum over $Y\\in\\mathcal{S}_X$ is equal to\n$\\mathscr{C}_X$ we see that $\\mathscr{N}_X$ becomes a full Hilbert graded\n$\\mathscr{C}_X$-module supported by $\\mathcal{S}_X$, cf.  Section \\ref{s:grad}. By\nTheorem \\ref{th:kghm} the imprimitivity $C^*$-algebra $\\mathcal{K}(\\mathscr{N}_X)$\nis equipped with a canonical $\\mathcal{S}_X$-graded structure.\n\nWe shall make a comment on $\\mathcal{K}(\\mathscr{M})$ in the more general the case\nwhen $\\mathscr{M}=\\oplus_i\\mathscr{M}_i$ is a direct sum of Hilbert $\\mathscr{A}$-modules\n$\\mathscr{M}_i$, cf. \\S\\ref{ss:gf}. First, it is clear that we have \n\\[\n\\mathcal{K}(\\mathscr{M})={\\textstyle\\sum^\\mathrm{c}_{ij}}\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i)\\cong\n(\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i))_{ij}.\n\\]\nNow assume that $\\mathcal{E},\\mathcal{E}_i$ are Hilbert spaces such that $\\mathscr{A}$ is a\n$C^*$-algebra of operators on $\\mathcal{E}$ and $\\mathscr{M}_i$ is a Hilbert\n$C^*$-submodule of $L(\\mathcal{E},\\mathcal{E}_i)$ such that\n$\\mathscr{A}_i\\equiv\\mathscr{M}_i^*\\cdot\\mathscr{M}_i$ is an ideal of $\\mathscr{A}$. \nThen by Proposition \\ref{pr:2ss} we have\n$\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i)\\cong\\mathscr{M}_i\\cdot\\mathscr{M}_j^*\\subset L(\\mathcal{E}_j,\\mathcal{E}_i)$. \n\nIn our case we take \n\\[\ni=Y\\in\\mathcal{S}_X,\\quad \\mathscr{M}_i=\\mathscr{C}_{YX}, \\quad \\mathscr{A}=\\mathscr{C}_X,\\quad\n\\mathcal{E}=\\mathcal{H}_X,\\quad  \\mathcal{E}_i=\\mathcal{H}_Y,\\quad   \\mathscr{A}_i=\\mathscr{C}_X^Y.\n\\]\nThen we get\n\\[\n\\mathcal{K}(\\mathscr{M}_j,\\mathscr{M}_i)\\equiv\\mathcal{K}(\\mathscr{C}_{ZX},\\mathscr{C}_{YX})\\cong\n\\mathscr{C}_{YX}\\cdot\\mathscr{C}_{ZX}^*=\\mathscr{C}_{YX}\\cdot\\mathscr{C}_{XZ}=\\mathscr{C}_{YZ}\n\\]\nby Proposition \\ref{pr:mxyz}.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\\begin{remark}\\label{re:squant}\n  We understood the role in our work of the imprimitivity algebra of\n  a Hilbert $C^*$-module thanks to a discussion with Georges\n  Skandalis: he recognized (a particular case of) the main\n  $C^*$-algebra $\\mathscr{C}$ we have constructed as the imprimitivity\n  algebra of a certain Hilbert $C^*$-module. Theorem \\ref{th:mor} is\n  a reformulation of his observation and of his abstract\n  construction of graded Hilbert $C^*$-modules in the present\n  framework (at the time of the discussion our definition of $\\mathscr{C}$\n  was rather different because we were working in a tensor product\n  formalism).  More generally, if $\\mathscr{M}$ is a full Hilbert\n  $\\mathscr{A}$-module then the imprimitivity $C^*$-algebra $\\mathcal{K}(\\mathscr{M})$ could\n  also be interpreted as Hamiltonian algebra of a system related in\n  some natural way to the initial one.  For example, this is a\n  natural method of ``second quantizing'' \\mbox{$N$-body} systems,\n  i.e.  introducing interactions which couple subsystems\n  corresponding to different cluster decompositions of the $N$-body\n  systems.  This is clear in the physical $N$-body situation\n  discussed in \\S\\ref{ss:cexample}\n\\end{remark}\n\n\n\\section{An intrinsic description}\n\\label{s:id}\n\\protect\\setcounter{equation}{0}\n\nWe begin with some preliminary facts on crossed products.  Let $X$\nbe a locally compact abelian group.  The next result, due to\nLandstad \\cite{Ld}, gives an ``intrinsic'' characterization of\ncrossed products of \\mbox{$X$-algebras} by the action of $X$.  We\nfollow the presentation from \\cite[Theorem 3.7]{GI4} which takes\nadvantage of the fact that $X$ is abelian.\n\n\\begin{theorem}\\label{th:land}\nA $C^*$-algebra $\\mathscr{A}\\subset \\mathscr{L}_X$ is a crossed product\nif and only for each $A\\in\\mathscr{A}$ we have:\n\\begin{itemize}\n\\vspace{-2mm} \n\\item\nif $k\\in X^*$ then $V_k^*AV_k\\in\\mathscr{A}$ and\n$\\lim_{k\\rarrow0}\\|V_k^*AV_k-A\\|=0$,\n\\item\nif $x\\in X$ then $U_xA\\in\\mathscr{A}$ and $\\lim_{x\\rarrow0}\\|(U_x-1)A\\|=0$.\n\\vspace{-2mm}\n\\end{itemize} \nIn this case one has $\\mathscr{A}=\\mathcal{A}\\rtimes X$ for a unique $X$-algebra\n$\\mathcal{A}\\subset\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$  and this algebra is given by\n\\begin{equation}\\label{eq:land}\n\\mathcal{A} =\\{\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)\\mid \n\\varphi(Q)S \\in {\\mathscr{A}} \\hspace{1mm}\\text{and}\\hspace{1mm}\n\\bar\\varphi(Q)S \\in {\\mathscr{A}}\n\\hspace{1mm}\\text{for all}\\hspace{1mm} S \\in \\mathscr{T}_X\\}.\n\\end{equation} \n\\end{theorem}\nNote that the second condition above is equivalent\nto $\\mathscr{T}_X\\cdot\\mathscr{A}=\\mathscr{A}$, cf. Lemma \\ref{lm:help}.\n\n\nThe following consequence of Landstad's theorem is an intrinsic \ndescription of $\\mathscr{C}_X(Y)$.\n\n\\begin{theorem}\\label{th:cxy} \n$\\mathscr{C}_X(Y)$ is the set of $A\\in \\mathscr{L}_X$ such that $U_y^*AU_y=A$ for all\n$y\\in Y$ and:\n\\begin{enumerate}\n\\item[{\\rm(1)}]\n$\\|U_x^*AU_x-A\\|\\to 0$ if $x\\to 0$ in $X$ and \n$\\|V_k^*AV_k-A\\|\\to 0$ if $k\\to 0$ in $X^*$,\n\\item[{\\rm(2)}]\n$\\|(U_x-1)A\\|\\to 0$ if $x\\to 0$ in $X$ and $\\|(V_k-1)A\\|\\to 0$ if\n$k\\to 0$ in $Y^\\perp$. \n\\end{enumerate}   \n\\end{theorem}\n\nBy ``$k\\to 0$ in $Y^\\perp$'' we mean: $k\\in Y^\\perp$ and $k\\to 0$.\nNote that the second condition above is equivalent to:\n\\begin{equation}\\label{eq:cxy}\n\\text{there are } \\theta\\in\\mathscr{T}_X,\\ \\psi\\in\\mathcal{C}_X(Y)\n\\text{ and }  B,C\\in\\mathscr{L}_X \\text{ such that } \nA=\\theta(P)B=\\psi(Q)C.\n\\end{equation}\nFor the proof, use $Y^\\perp\\cong (X\/Y)^*$ and apply Lemma\n\\ref{lm:help}. In particular, the last factorization shows that for\neach $\\varepsilon>$ there is a compact set $M\\subset X$ such that\n$\\|\\cchi_V(Q)A\\|<\\varepsilon$, where $V=X\\setminus(M+Y)$.\n\n\\noindent{\\bf Proof of Theorem \\ref{th:cxy}:} Let $\\mathscr{A}\\subset \\mathscr{L}_X$\nbe the set of operators $A$ satisfying the conditions from the\nstatement of the theorem.  We first prove that $\\mathscr{A}$ satisfies the\ntwo conditions of Theorem \\ref{th:land}. Let $A\\in\\mathscr{A}$.  We have to\nshow that $A_p\\equiv V_p^*AV_p\\in\\mathscr{A}$ and $\\|V_p^*AV_p-A\\|\\to0$ as\n$p\\to0$.  From the commutation relations $U_xV_p=p(x)V_pU_x$ we get\n$\\|(U_x-1)A_p\\|=\\|(U_x-p(x))A\\|\\to0$ if $x\\to0$ and the second part\nof condition 1 of the theorem is obviously satisfied by $A_p$. Then\nfor $y\\in Y$\n$$\nU_y^*A_pU_y=U_y^*V_p^*AV_pU_y=V_p^*U_y^*AU_yV_p=V_p^*AV_p=A_p.\n$$ Condition 2 is clear so we have $A_p\\in\\mathscr{A}$ and the fact that\n$\\|V_p^*AV_p-A\\|\\to0$ as $p\\to0$ is obvious. That $A$ satisfies the\nsecond Landstad condition, namely that for each $a\\in X$ we have\n$U_aA\\in\\mathscr{A}$ and $\\|(U_a-1)A\\|\\to0$ as $a\\to0$, is also clear because\n$\\|[U_a,V_k]\\|\\to0$ as $k\\to0$.\n\nNow we have to find the algebra $\\mathcal{A}$ defined by \\eqref{eq:land}.\nAssume that $\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ satisfies $\\varphi(Q)S\\in\\mathscr{A}$ for all\n$S\\in \\mathscr{T}_X$. Since $U_y^*\\varphi(Q)U_y=\\varphi(Q-y)$ we get\n$(\\varphi(Q)-\\varphi(Q-y))S=0$ for all such $S$ and all $y\\in Y$,\nhence $\\varphi(Q)-\\varphi(Q-y)=0$ which means $\\varphi\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X\/Y)$.\nWe shall prove that $\\varphi\\in\\mathcal{C}_X(Y)$ by reductio ad absurdum. \n\nIf $\\varphi\\notin\\mathcal{C}_X(Y)$ then there is $\\mu>0$ and there is a\nsequence of points $x_n\\in X$ such that $x_n\/Y\\to\\infty$ and\n$|\\varphi(x_n)|>2\\mu$. From the uniform continuity of $\\varphi$ we\nsee that there is a compact neighborhood $K$ of zero in $X$ such\nthat $|\\varphi|>\\mu$ on $\\bigcup_n(x_n+K)$.  Let $K'$ be a compact\nneighborhood of zero such that $K'+K'\\subset K$ and let us choose\ntwo positive not zero functions $\\psi,f\\in\\cc_{\\mathrm{c}}(K')$. We define $S\\in\n\\mathscr{T}_X$ by $Su=\\psi*u$ and recall that $\\mbox{\\rm supp\\! } Su\\subset\\mbox{\\rm supp\\! }\n\\psi+\\mbox{\\rm supp\\! } u$. Thus $\\mbox{\\rm supp\\! } SU_{x_n}^*f\\subset K'+x_n+K'\\subset\nx_n+K$. Now let $V$ be as in the remarks after \\eqref{eq:cxy}. Since\n$\\pi_Y(x_n)\\to\\infty$ we have $x_n+K\\subset V$ for $n$ large enough,\nhence\n$$\n\\|\\cchi_V(Q)\\varphi(Q)SU_{x_n}^*f\\|\\geq\\mu\\|SU_{x_n}^*f\\|=\n\\mu\\|Sf\\| >0.\n$$\nOn the other hand, for each $\\varepsilon>0$ one can choose $V$ such\nthat $\\|\\cchi_V(Q)\\varphi(Q)S\\|<\\varepsilon$.  Then we shall have\n$\\|\\cchi_V(Q)\\varphi(Q)SU_{x_n}^*f\\|\\leq\\varepsilon\\|f\\|$ so\n$\\mu\\|Sf\\|\\leq\\varepsilon\\|f\\|$ for all $\\varepsilon>0$ which is\nabsurd.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nWe now give a similar characterization of $\\mathscr{C}_{XY}(Z)$ where $X,Y$\nis a compatible pair of closed subgroups of an lca group $G$.\n\n\\begin{theorem}\\label{th:yzintr}\n  $\\mathscr{C}_{XY}(Z)$ is the set of $T\\in\\mathscr{L}_{XY}$ satisfying the\n  following conditions:\n\\begin{enumerate} \\vspace{-2mm}\n\\item[{\\rm(1)}] $U_z^*T U_z=T$ if $z\\in Z$ and $\\|V^*_k T V_k-T\\|\\to\n  0$ if $k\\to 0$ in $(X+Y)^*$\n\\item[{\\rm(2)}]\n$\\|(U_x-1)T\\|\\to 0$ if $x\\to 0$ in $X$ and \n$\\|T(U_y-1)\\|\\to 0$ if $y\\to 0$ in $Y$, \n\\item[{\\rm(3)}]\n$\\|(V_k-1)T\\|\\to 0$ if $k\\to 0$ in $(X\/Z)^*$ and \n$\\|T(V_k-1)\\|\\to 0$ if $k\\to 0$ in $(Y\/Z)^*$.\n\\end{enumerate}\n\\end{theorem}\n\nBefore the proof we make some preliminary comments. We think of\n$X+Y$ as a closed subgroup of $G\\in\\mathscr{S}$ which contains $X$ and $Y$\nas closed subgroups.  Each character $k\\in(X+Y)^*$ defines by\nrestriction a character $k|_X\\in X^*$ and the map $k\\mapsto k|_X$ is\na continuous open surjection. And similarly if $X$ is replaced by\n$Y$. In (1) the operator $V_k$ acts in $L^2(X)$ as multiplication by\n$k|_X$ and in $L^2(Y)$ as multiplication by $k|_Y$. In the first\npart of (3) we take $k\\in X^*$ and identify $(X\/Z)^*$ with the\northogonal of $Z$ in $X^*$ and similarly for the second part.\n\nAssumptions (2) and (3) of Theorem \\ref{th:yzintr} are decay\nconditions in certain directions in $P$ and $Q$ space. Indeed, by\nLemma \\ref{lm:help} condition (2) is equivalent to:\n\\begin{equation}\\label{eq:cond1}\n\\text{there are } S_1\\in \\mathscr{T}_X, S_2\\in \\mathscr{T}_Y \\text{ and }\nR_1,R_2\\in\\mathscr{L}_{XY} \\text{ such that } T=S_1R_1=R_2S_2.\n\\end{equation}\nRecall that $\\mathscr{T}_X\\cong\\cc_{\\mathrm{o}}(X^*)$ for example. Then\ncondition (3) is equivalent to:\n\\begin{equation}\\label{eq:cond2}\n\\text{there are } S_1\\in \\mathcal{C}_X(Z), S_2\\in \\mathcal{C}_Y(Z) \\text{ and }\nR_1,R_2\\in\\mathscr{L}_{XY} \\text{ such that } T=S_1R_1=R_2S_2.\n\\end{equation}\n\n\n\\noindent{\\bf Proof of Theorem \\ref{th:yzintr}:} The set $\\mathscr{C}$ of all\nthe operators satisfying the conditions of the theorem is clearly a\nclosed subspace of $\\mathscr{L}_{XY}$.  We have $\\mathscr{C}_{X,Y}(Z)\\subset\\mathscr{C}$\nbecause \\eqref{eq:cond1}, \\eqref{eq:cond2} are satisfied by any\n$T\\in\\mathscr{C}_{XY}(Z)$ as a consequence of Theorem \\ref{th:nxyz}.  Then we\nget:\n$$\n\\mathscr{C}_Y(Z)= \n\\mathscr{C}_{XY}^*(Z)\\cdot\\mathscr{C}_{XY}(Z)\\subset \\mathscr{C}^*\\cdot\\mathscr{C}, \n\\hspace{1mm}\n\\mathscr{C}_X(Z)=\n\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{XY}^*(Z)\\subset \\mathscr{C}\\cdot\\mathscr{C}^*.\n$$ \nWe prove that equality holds in both these relations. We show, for\nexample, that $A\\equiv TT^*$ belongs to $\\mathscr{C}_X(Z)$ if $T\\in\\mathscr{C}$ and\nfor this we shall use Theorem \\ref{th:cxy} with $Y$ replaced by\n$Z$. That $U_z^*AU_z=A$ for $z\\in Z$ is clear. From \\eqref{eq:cond1}\nwe get $A=S_1R_1R_1^*S_1^*$ with $S_1\\in\\mathscr{T}_X$ hence\n$\\|(U_x-1)A\\|\\to 0$ and $\\|A(U_x-1)\\|\\to 0$ as $x\\to0$ in $X$ are\nobvious and imply $\\|U_x^*AU_x-A\\|\\to 0$. Then \\eqref{eq:cond2}\nimplies $A=\\psi(Q)C$ with $\\psi\\in\\mathcal{C}_X(Z)$ and bounded $C$ hence\n\\eqref{eq:cxy} is satisfied.\n\nThat $\\mathscr{C}\\rc_Y(Z)\\subset\\mathscr{C}$ is easily proven because $T=SA$ has the\nproperties \\eqref{eq:cond1} and \\eqref{eq:cond2} if $S$ belongs to\n$\\mathscr{C}$ and $A$ to $\\mathscr{C}_Y(Z)$, cf. Theorem \\ref{th:cxy}. From what we\nhave shown above we get $\\mathscr{C}\\rc^*\\mathscr{C}\\subset\\mathscr{C}\\rc_Y(Z)\\subset\\mathscr{C}$ so\n$\\mathscr{C}$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$. On the other hand,\n$\\mathscr{C}_{XY}(Z)$ is a Hilbert $C^*$-submodule of $\\mathscr{L}_{XY}$ such that\n$\\mathscr{C}_{XY}^*(Z)\\cdot\\mathscr{C}_{XY}(Z)=\\mathscr{C}^*\\cdot\\mathscr{C}$ and\n$\\mathscr{C}_{XY}(Z)\\cdot\\mathscr{C}_{XY}^*(Z)=\\mathscr{C}\\cdot\\mathscr{C}^*$. Since\n$\\mathscr{C}_{XY}(Z)\\subset\\mathscr{C}$ we get $\\mathscr{C}=\\mathscr{C}_{XY}(Z)$ from Proposition\n\\ref{pr:clsubmod}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\nIf $Z=X\\cap Y$ then Theorem \\ref{th:yzintr} gives an intrinsic\ndescription of the space $\\mathscr{T}_{XY}$. For example:\n\n\\begin{corollary}\\label{co:txyintr}\nIf $X\\supset Y$ then $\\mathscr{T}_{XY}$ is the set of $T\\in\\mathscr{L}_{XY}$\nsatisfying $U_y^*T U_y=T$    if $y\\in Y$ and such that:\n$U_xT\\to T$ if $x\\to 0$ in $X$, \n$V^*_k T V_k\\to T$ if $k\\to 0$ in $X^*$ and \n$V_kT\\to T$ if $k\\to 0$ in $Y^\\perp$.\n\\end{corollary}\n\n\nIn the rest of this section we describe the structure of the objects\nintroduced in Section \\ref{s:grass} when the subgroups are\ncomplemented, e.g. if $\\mathcal{S}$ consists of finite dimensional vector\nspaces.\n\n\nWe say that \\emph{$Z$ is complemented in $X$} if $X=Z\\oplus E$ for\nsome closed subgroup $E$ of $X$. If $X,Z$ are equipped with Haar\nmeasures then $X\/Z$ is equipped with the quotient Haar measure and we\nhave $E\\simeq X\/Z$.  If $Z$ is complemented in $X$ and $Y$ then\n$\\mathscr{C}_{XY}(Z)$ can be expressed as a tensor product.\n\n\\begin{proposition}\\label{pr:def3}\nIf $Z$ is complemented in $X$ and $Y$ then\n\\begin{equation}\\label{eq:ryzsum}\n\\mathscr{C}_{XY}(Z)\\simeq \\mathscr{T}_Z\\otimes \\mathscr{K}_{X\/Z,Y\/Z}.\n\\end{equation}\nIf $Y\\subset X$ then $\\mathscr{T}_{XY}\\simeq \\mathscr{T}_Y\\otimes L^2(X\/Y)$ tensor\nproduct of Hilbert $C^*$-modules.\n\\end{proposition} \n\\proof Note first that the tensor product in \\eqref{eq:ryzsum} is\ninterpreted as the exterior tensor product of the Hilbert\n$C^*$-modules $\\mathscr{T}_Z$ and $\\mathscr{K}_{X\/Z,Y\/Z}$. Let $X=Z\\oplus E$ and\n$Y=Z\\oplus F$ for some closed subgroups $E,F$. Then, as explained in\n\\S\\ref{ss:ha}, we may also view the tensor product as the norm closure\nin the space of continuous operators from $L^2(Y)\\simeq L^2(Z)\\otimes\nL^2(F)$ to $L^2(X)\\simeq L^2(Z)\\otimes L^2(E)$ of the linear space\ngenerated by the operators of the form $T\\otimes K$ with $T\\in\n\\mathscr{T}_Z$ and $K\\in \\mathscr{K}_{EF}$.\n\n\n\n\nWe now show that under the conditions of the proposition $X+Y\\simeq\nZ\\oplus E\\oplus F$ algebraically and topologically. The natural map\n$\\theta:Z\\oplus E\\oplus F\\to Z+E+F=X+Y$ is a continuous bijective\nmorphism, we have to prove that it is open.  Since $X,Y$ are\ncompatible, the  map \\eqref{eq:nat} is a continuous open\nsurjection.  If we represent $X\\oplus Y\\simeq Z\\oplus Z\\oplus E\\oplus\nF$ then this map becomes $\\phi(a,b,c,d)=(a-b)+c+d$.  Let\n$\\psi=\\xi\\oplus{\\rm id}_E\\oplus{\\rm id}_F$ where $\\xi:Z\\oplus Z\\to Z$\nis given by $\\xi(a,b)=a-b$. Then $\\xi$ is continuous surjective and\nopen because if $U$ is an open neighborhood of zero in $Z$ then\n$U-U$ is also an open neighborhood of zero. Thus\n$\\psi:(Z\\oplus Z)\\oplus E\\oplus F \\to Z\\oplus E\\oplus F$ is a\ncontinuous open surjection and $\\phi=\\theta\\circ\\psi$. So if $V$ is\nopen in $Z\\oplus E\\oplus F$ then there is an open \n$U\\subset Z\\oplus Z\\oplus E\\oplus F$ such that $V=\\psi(U)$ and then\n$\\theta(V)=\\theta\\circ\\psi(U)=\\phi(U)$ is open in $Z+E+F$.\n\nThus we may identify $L^2(Y)\\simeq L^2(Z)\\otimes L^2(F)$ and\n$L^2(X)\\simeq L^2(Z)\\otimes L^2(E)$ and we must describe the norm\nclosure of the set of operators $T_{XY}(\\varphi)\\psi(Q)$ with\n$\\varphi\\in\\cc_{\\mathrm{c}}(X+Y)$ (cf. the remark after \\eqref{eq:ryz} and the fact\nthat $X+Y$ is closed) and $\\psi\\in\\cc_{\\mathrm{o}}(Y\/Z)$. Since $X+Y\\simeq Z\\oplus\nE\\oplus F$ and $Y=Z\\oplus F$ it suffices to describe the clspan of the\noperators $T_{XY}(\\varphi)\\psi(Q)$ with\n$\\varphi=\\varphi_Z\\otimes\\varphi_E\\otimes\\varphi_F$ and\n$\\varphi_Z,\\varphi_E,\\varphi_F$ continuous functions with compact\nsupport on $Z,E,F$ respectively and $\\psi=1\\otimes\\eta$ where $1$ is\nthe function identically equal to $1$ on $Z$ and\n$\\eta\\in\\cc_{\\mathrm{o}}(F)$. Then, if $x=(a,c)\\in Z\\times E$ and $y=(b,d)\\in\nZ\\times F$, we get:\n$$\n(T_{XY}(\\varphi)\\psi(Q)u)(a,c)=\\int_{Z\\times F} \\varphi_Z(a-b)\n\\varphi_E(c)\n\\varphi_F(d)  \\eta(d) u(b,d) \\text{d} b \\text{d} d. \n$$ But this is just\n$C(\\varphi_Z)\\otimes\\ket{\\varphi_E}\\bra{\\bar\\eta\\bar\\varphi_F}$ where\n$\\ket{\\varphi_E}\\bra{\\bar\\eta\\bar\\varphi_F}$ is a rank one operator\n$L^2(F)\\to L^2(E)$ and $C(\\varphi_Z)$ is the operator of convolution\nby $\\varphi_Z$ on $L^2(Z)$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nIf $X\\cap Y$ is complemented in $X$ and $Y$ then $\\mathscr{C}_{XY}$ can be\nexpressed (non canonically) as a tensor product.\n\n\\begin{proposition}\\label{pr:xytens}\nIf $X\\cap Y$ is complemented in $X$ and $Y$ then\n\\[\n\\mathscr{C}_{XY}\\simeq \\mathscr{C}_{X\\cap Y} \\otimes \\mathscr{K}_{X\/Y,Y\/X}.\n\\] \nIn particular,\nif $X\\supset Y$ then $\\mathscr{C}_{XY}\\simeq \\mathscr{C}_{Y} \\otimes \\mathcal{H}_{X\/Y}$.\n\\end{proposition}\n\\proof If $X=(X\\cap Y)\\oplus E$ and $Y=(X\\cap Y)\\oplus F$ then we have\nto show that $\\mathscr{C}_{XY}\\simeq \\mathscr{C}_{X\\cap Y} \\otimes \\mathscr{K}_{EF}$ where the\ntensor product may be interpreted either as the exterior tensor\nproduct of the Hilbert $C^*$-modules $\\mathscr{C}_{X\\cap Y}$ and $\\mathscr{K}_{EF}$ or\nas the norm closure in the space of continuous operators from\n$L^2(Y)\\simeq L^2(X\\cap Y)\\otimes L^2(F)$ to $L^2(X)\\simeq L^2(X\\cap\nY)\\otimes L^2(E)$ of the algebraic tensor product of $\\mathscr{C}_{X\\cap Y}$\nand $\\mathscr{K}_{EF}$.  From Proposition \\ref{pr:def3} with $Z=X\\cap Y$ we\nget $\\mathscr{T}_{XY}\\simeq\\mathscr{T}_{X\\cap Y}\\otimes\\mathscr{K}_{EF}$. The relations\n\\eqref{eq:main} and the Definition \\ref{df:main} imply\n$\\mathscr{C}_{XY}=\\mathscr{T}_{XY}\\cdot\\mathcal{C}_Y^{X\\cap Y}$ and we clearly have\n\\[\n\\mathcal{C}_Y^{X\\cap Y}={\\textstyle\\sum_{Z\\subset X\\cap Y}^\\mathrm{c}} \\mathcal{C}_Y(Z)\n\\simeq {\\textstyle\\sum_{Z\\subset X\\cap Y}^\\mathrm{c}} \n\\mathcal{C}_{X\\cap Y}(Z)\\otimes \\cc_{\\mathrm{o}}(F)\n\\simeq \\mathcal{C}_{X\\cap Y}\\otimes \\cc_{\\mathrm{o}}(F).\n\\]     \nThen we get\n\\[\n\\mathscr{C}_{XY}\\simeq \\mathscr{T}_{X\\cap Y}\\otimes\\mathscr{K}_{EF}\\cdot \n\\mathcal{C}_{X\\cap Y}\\otimes \\cc_{\\mathrm{o}}(F)=\n\\big(\\mathscr{T}_{X\\cap Y}\\cdot\\mathcal{C}_{X\\cap Y} \\big)\\otimes\n\\big(\\mathscr{K}_{EF}\\cdot\\cc_{\\mathrm{o}}(F)\\big)\n\\]\nand this is $\\mathscr{C}_{X\\cap Y} \\otimes \\mathscr{K}_{EF}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nIf $Z$ is complemented in $X$ and $Y$ then Theorem \\ref{th:yzintr}\ncan be improved. We shall describe this improvement only in the\nEuclidean case which will be useful in our treatment of\nnonrelativistic Hamiltonians. Thus below we assume that $X,Y$ are\nsubspaces of an Euclidean space (see \\S\\ref{ss:mouint} for\nnotations). Note that $V_k$ is the operator of multiplication by the\nfunction $x\\mapsto\\mathrm{e}^{i\\braket{x}{k}}$ where the scalar product\n$\\braket{x}{k}$ is well defined for any $x,k$ in the ambient space\n$\\mathcal{X}$. \n\n\\begin{theorem}\\label{th:xyzeintr}\n$\\mathscr{C}_{XY}(Z)$ is the set of $T\\in\\mathscr{L}_{XY}$ satisfying:\n\\begin{enumerate} \n\\item[{\\rm(1)}] \n$U_z^*T U_z=T$ for $z\\in Z$ and\n$\\|V^*_z T V_z-T\\|\\to 0$ if $z\\to 0$ in $Z$,\n\\item[{\\rm(2)}] \n$\\|(U_x-1)T\\|\\to 0$ if $x\\to 0$ in $X$ and $\\|(V_k-1)T\\|\\to 0$ if\n$k\\to 0$ in $X\/Z$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\\label{re:xyzeintr}\nCondition 2 may be replaced by\n\\begin{compactenum}\n\\item[{\\rm(2$'$)}]\n$\\|T(U_y-1)\\|\\to 0$ if $y\\to 0$ in $Y$ and $\\|T(V_k-1)\\|\\to 0$ if\n$k\\to 0$ in $Y\/Z$.\n\\end{compactenum}\nThis will be clear from the next proof.\n\\end{remark}\n\\proof Let $\\mathcal{F}\\equiv\\mathcal{F}_Z$ be the Fourier transformation in the\nspace $Z$, this is a unitary operator in the space $L^2(Z)$ which\ninterchanges the position and momentum observables $Q_Z,P_Z$. We\ndenote also by $\\mathcal{F}$ the operators $\\mathcal{F}\\otimes1_{\\mathcal{H}_{X\/Z}}$ and\n$\\mathcal{F}\\otimes1_{\\mathcal{H}_{Y\/Z}}$ which are unitary operators in the spaces\n$\\mathcal{H}_X$ and $\\mathcal{H}_Y$ due to \\eqref{eq:xyzint}.  If $S=\\mathcal{F} T\n\\mathcal{F}^{-1}$ then $S$ satisfies the following conditions:\n\\begin{enumerate} \n\\item[(i)]\n$V_z^*S V_z=S$ for $z\\in Z$, $\\|(V_z-1)S\\|\\to 0$ if $z\\to 0$ in $Z$,\nand $\\|U_z S U^*_z-S\\|\\to 0$ if $z\\to 0$ in $Z$;\n\\item[(ii)] \n$\\|(U_x-1)S\\|\\to 0$ and $\\|(V_x-1)S\\|\\to 0$ if $x\\to 0$ in $X\/Z$.\n\\end{enumerate}\nFor the proof, observe that the first part of condition (2) may be\nwritten as the conjunction of the two relations $\\|(U_z-1)T\\|\\to 0$\nif $z\\to 0$ in $Z$ and $\\|(U_x-1)T\\|\\to 0$ if $x\\to 0$ in $X\/Z$.  We\nshall work in the representations\n\\begin{equation}\\label{eq:fiber}\n\\mathcal{H}_X= L^2(Z;\\mathcal{H}_{X\/Z}) \\quad \\text{and} \\quad\n\\mathcal{H}_Y= L^2(Z;\\mathcal{H}_{Y\/Z}).\n\\end{equation} \nFrom the relation $V_z^*S V_z=S$ for all $z\\in Z$ it follows that\nthere is a bounded weakly measurable function\n$S(\\cdot):Z\\to\\mathscr{L}_{X\/Z,Y\/Z}$ such that in the representations\n\\eqref{eq:fiber} $S$ is the operator of multiplication by\n$S(\\cdot)$. Then $\\|U_z S U^*_z-S\\|\\to 0$ if $z\\to 0$ in $Z$ means\nthat the function $S(\\cdot)$ is uniformly continuous. And clearly\n$\\|(V_z-1)S\\|\\to 0$ if $z\\to 0$ in $Z$ is equivalent to the fact\nthat $S(\\cdot)$ tends to zero at infinity. Thus we see that\n$S(\\cdot)\\in\\cc_{\\mathrm{o}}(Z;\\mathscr{L}_{X\/Z,Y\/Z})$.\nThe condition (ii) can now be written\n\\[\n\\sup_{z\\in Z}\\big(\\|(U_x-1)S(z)\\|+ \\|(V_x-1)S(z)\\|\\big)\\to 0\n\\quad \\text{if } x\\to 0 \\text{ in } X\/Z.\n\\]\nFrom the Riesz-Kolmogorov theorem it follows that each $S(z)$ is a\ncompact operator. Thus we have $S(\\cdot)\\in\\cc_{\\mathrm{o}}(Z;\\mathscr{K}_{X\/Z,Y\/Z})$\nwhich implies $T\\in\\mathscr{C}_{XY}(Z)$ by Proposition \\ref{pr:def3}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\\begin{remark}\\label{re:half}\nSince $S(\\cdot)$ is continuous and tends to zero at infinity, for\neach $\\varepsilon>0$ there are points $z_1,\\dots,z_n\\in Z$ and\ncomplex functions $\\varphi_1,\\dots,\\varphi_n\\in\\cc_{\\mathrm{c}}(Z)$ such that\n\\[\n\\|S(z)-{\\textstyle\\sum_k}\\varphi_k(z)S(z_k)\\|\\leq\\varepsilon\\quad\n\\forall z\\in Z.\n\\]\nThe operators $S(z_k)$ being compact, applying once again the\nRiesz-Kolmogorov theorem we get\n\\[\n\\sup_{z\\in Z}\\big(\\|S(z)(U_y-1)\\|+ \\|S(z)(V_y-1)\\|\\big)\\to 0 \n\\quad \\text{if } y\\to 0 \\text{ in } Y\/Z.\n\\]\nThis explains why the second parts of conditions (2) and (3) of\nTheorem \\ref{th:yzintr} is not needed. \n\\end{remark}\n\n\n\n\n\\section{Affiliated operators}\n\\label{s:af}\n\\protect\\setcounter{equation}{0}\n\nIn this section we give examples of self-adjoint operators\naffiliated to the algebra $\\mathscr{C}$ constructed in Section \\ref{s:grass}\nand then we give a formula for their essential spectrum. We refer to\n\\S\\ref{ss:grca} for terminology and basic results related to the\nnotion of affiliation that we use and to \\cite{ABG,GI1,DG3} for\ndetails.\n\nWe recall that a self-adjoint operator $H$ on a Hilbert space $\\mathcal{H}$\nis \\emph{strictly affiliated} to a $C^*$-algebra of operators $\\mathscr{A}$\non $\\mathcal{H}$ if $(H+i)^{-1}\\in\\mathscr{A}$ (then $\\varphi(H)\\in\\mathscr{A}$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$) and if $\\mathscr{A}$ is the clspan of the elements\n$\\varphi(H)A$ with $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$ and $A\\in\\mathscr{A}$. This class\nof operators has the advantage that each time $\\mathscr{A}$ is\nnon-degenerately represented on a Hilbert space $\\mathcal{H}'$ with the help\nof a morphism $\\mathscr{P}:\\mathscr{A}\\to L(\\mathcal{H}')$, the observable $\\mathscr{P} H$ is\nrepresented by a usual (densely defined) self-adjoint operator on\n$\\mathcal{H}'$.\n\nThe diagonal algebra\n\\begin{equation}\\label{eq:d}\n\\mathscr{T}_{\\text{d}}\\equiv(\\mathscr{T}_{\\mathcal{S}})_\\text{d}=\\oplus_{X\\in\\mathcal{S}} \\mathscr{T}_X\n\\end{equation}\nhas a simple physical interpretation: this is the $C^*$-algebra\ngenerated by the kinetic energy operators.  Since\n$\\mathscr{C}_{XX}=\\mathscr{C}_X\\supset\\mathscr{C}_{X}(X)= \\mathscr{T}_X$ we see that $\\mathscr{T}_\\text{d}$\nis a $C^*$-subalgebra of $\\mathscr{C}$. From \\eqref{eq:nxyz},\n\\eqref{eq:txy1}, \\eqref{eq:txy2} and the Cohen-Hewitt theorem we get\n\\begin{equation}\\label{eq:ed}\n\\mathscr{C}(Z)\\mathscr{T}_\\text{d}=\\mathscr{T}_\\text{d}\\mathscr{C}(Z)=\\mathscr{C}(Z)\\quad \\forall Z\\in\\mathcal{S} \\quad \n\\text{and}\\hspace{2mm} \\mathscr{C} \\mathscr{T}_\\text{d}=\\mathscr{T}_\\text{d}\\mathscr{C}=\\mathscr{C}.\n\\end{equation} \nIn other terms, $\\mathscr{T}_\\text{d}$ acts\nnon-degenerately\\symbolfootnote[2]{\\ Note that if $\\mathcal{S}$ has a\n  largest element $\\mathcal{X}$ then the algebra $\\mathscr{C}(\\mathcal{X})$ acts on each\n  $\\mathscr{C}(Z)$ but this action is degenerate.} \non each $\\mathscr{C}(Z)$ and on $\\mathscr{C}$. It follows that a self-adjoint\noperator strictly affiliated to $\\mathscr{T}_\\text{d}$ is also strictly\naffiliated to $\\mathscr{C}$. \n\n\nFor each $X\\in\\mathcal{S}$ let $h_X:X^*\\to\\mathbb{R}$ be a continuous function\nsuch that $|h_X(k)|\\to\\infty$ if $k\\to\\infty$ in $X^*$. Then the\nself-adjoint operator $K_X\\equiv h_X(P)$ on $\\mathcal{H}_X$ is strictly\naffiliated to $\\mathscr{T}_X$ and the norm of $(K_X+i)^{-1}$ is equal to\n$\\sup_k(h^2_X(k)+1)^{-1\/2}$. Let $K\\equiv\\bigoplus_{X\\in\\mathcal{S}}K_X$,\nthis is a self-adjoint operator $\\mathcal{H}$. Clearly $K$ is affiliated to \n$\\mathscr{T}_\\text{d}$ if and only if \n\\begin{equation}\\label{eq:kin}\n\\lim_{X\\to\\infty}\\sup\\nolimits_{k}(h^2_X(k)+1)^{-1\/2}=0\n\\end{equation}\nand then $K$ is strictly affiliated to $\\mathscr{T}_\\text{d}$ (the set $\\mathcal{S}$ is\nequipped with the discrete topology). If the functions $h_X$ are\npositive this means that $\\min h_X$ tends to infinity when\n$X\\to\\infty$. One could avoid such a condition by considering an\nalgebra larger then $\\mathscr{C}$ such as to contain \n$\\prod_{X\\in\\mathcal{S}} \\mathscr{T}_X$, but we shall not develop this here.\n\nNow let $H=K+I$ with $I\\in\\mathscr{C}$ (or in the multiplier algebra) a\nsymmetric element. Then\n\\begin{equation}\\label{eq:res}\n(\\lambda-H)^{-1}=(\\lambda-K)^{-1}\\left(1-I(\\lambda-K)^{-1}\\right)^{-1}\n\\end{equation}\nif $\\lambda\\notin\\mathrm{Sp}(H)\\cup\\mathrm{Sp}(K)$ . Thus $H$ is strictly affiliated\nto $\\mathscr{C}$.  We interpret $H$ as the Hamiltonian of our system of\nparticles when the kinetic energy is $K$ and the interactions\nbetween particles are described by $I$. Even in the simple case\n$I\\in\\mathscr{C}$ these interactions are of a very general nature being a\nmixture of $N$-body and quantum field type interactions (which\ninvolve creation and annihilation operators so the number of\nparticles is not preserved).\n\nWe shall now use Theorem \\ref{th:gas} in order to compute the\nessential spectrum of an operator like $H$. The case of unbounded\ninteractions will be treated later on. Let $\\mathscr{C}_{\\geq E}$ be the\n$C^*$-subalgebra of $\\mathscr{C}$ determined by $E\\in\\mathcal{S}$ according to the\nrules of $\\S\\ref{ss:grca}$. More explicitly, we set\n\\begin{equation}\\label{eq:geqe}\n\\mathscr{C}_{\\geq E}={\\textstyle\\sum^\\mathrm{c}_{F\\supset E}\\mathscr{C}(F)}\\cong \n\\big({\\textstyle\\sum^\\mathrm{c}_{F\\supset E}}\n\\mathscr{C}_{XY}(F)\\big)_{X\\cap Y\\supset E}\n\\end{equation}\nand note that $\\mathscr{C}_{\\geq E}$ lives on the subspace $\\mathcal{H}_{\\geq\n  E}=\\bigoplus_{X\\supset E}\\mathcal{H}_X$ of $\\mathcal{H}$. Since in the second sum\nfrom \\eqref{eq:geqe} the group $F$ is such that $E\\subset F\\subset\nX\\cap Y$ the algebra $\\mathscr{C}_{\\geq E}$ is strictly included in the\nalgebra $\\mathscr{C}_\\mathcal{T}$ obtained by taking $\\mathcal{T}=\\{F\\in\\mathcal{S} \\mid F\\supset\nE\\}$ in \\eqref{eq:t}. \n\nLet $\\mathscr{P}_{\\geq E}$ be the canonical idempotent morphism of $\\mathscr{C}$\nonto $\\mathscr{C}_{\\geq E}$ introduced in \\S\\ref{ss:grca}. We consider\nthe self-adjoint operator on the Hilbert space $\\mathcal{H}_{\\geq E}$\ndefined as follows:\n\\begin{equation}\\label{eq:post}\n H_{\\geq E}=K_{\\geq E}+I_{\\geq E} \\quad \\text{where} \\quad \nK_{\\geq E}= \\oplus_{X\\geq E} K_X \\hspace{2mm} \\text{and} \n\\hspace{2mm} I_{\\geq E}=\\mathscr{P}_{\\geq E}I.\n\\end{equation}\nThen $H_{\\geq E}$ is strictly affiliated to $\\mathscr{C}_{\\geq E}$ and it\nfollows easily from \\eqref{eq:res} that\n\\begin{equation}\\label{eq:pre}\n\\mathscr{P}_{\\geq E}\\varphi(H)=\\varphi(H_{\\geq E}) \\quad \\forall\n\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R}).\n\\end{equation} \nNow let us assume that the group $O=\\{0\\}$ belongs to $\\mathcal{S}$. Then we\nhave\n\\begin{equation}\\label{eq:O}\n\\mathscr{C}(O)=K(\\mathcal{H}).\n\\end{equation} \nIndeed, from \\eqref{eq:nxyz} we get\n$\\mathscr{C}_{XY}(O)=\\mathscr{T}_{XY}\\cdot\\cc_{\\mathrm{o}}(Y)=\\mathscr{K}_{XY}$ which implies the\npreceding relation. If we also assume that $\\mathcal{S}$ is atomic and we\ndenote $\\mathcal{P}(\\mathcal{S})$ its set of atoms, then from Theorem \\ref{th:ga} we\nget a canonical embedding\n\\begin{equation}\\label{eq:quotc}\n\\mathscr{C}\/K(\\mathcal{H})\\subset\\displaystyle\\mbox{$\\textstyle\\prod$}_{\\substack{i}}\\nolimits_{E\\in\\mathcal{P}(\\mathcal{S})}\\mathscr{C}_{\\geq E}\n\\end{equation} \ndefined by the morphism $\\mathscr{P}\\equiv(\\mathscr{P}_{\\geq E})_{E\\in\\mathcal{P}(\\mathcal{S})}$.\nThen from \\eqref{eq:es2} we obtain:\n\\begin{equation}\\label{eq:ess1}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{E\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq E}).\n\\end{equation}\nOur next purpose is to prove a similar formula for a certain class\nof unbounded interactions $I$.\n\nLet $\\mathcal{G}\\equiv\\mathcal{G}_\\mathcal{S}=D(|K|^{1\/2})$ be the form domain of $K$\nequipped with the graph topology. Then $\\mathcal{G}\\subset\\mathcal{H}$ continuously\nand densely so after the Riesz identification of $\\mathcal{H}$ with its\nadjoint space $\\mathcal{H}^*$ we get the usual scale\n$\\mathcal{G}\\subset\\mathcal{H}\\subset\\mathcal{G}^*$ with continuous and dense embeddings.\nLet us denote\n\\begin{equation}\\label{eq:jap}\n\\jap{K} =|K+i|=\\sqrt{K^2+1}.\n\\end{equation}\nThen $\\jap{K}^{1\/2}$ is a self-adjoint operator on $\\mathcal{H}$ with domain\n$\\mathcal{G}$ and $\\jap{K}$ induces an isomorphism $\\mathcal{G}\\to\\mathcal{G}^*$.  The\nfollowing result is a straightforward consequence of Theorem 2.8 and\nLemma 2.9 from \\cite{DG3}. \n\n\\begin{theorem}\\label{th:af}\nLet $I:\\mathcal{G}\\to\\mathcal{G}^*$ be a continuous symmetric operator and let us\nassume that there are real numbers $\\mu,a$ with $0<\\mu<1$ such that\none of the following conditions is satisfied:\n\\begin{compactenum}[(i)]\n\\item\n$\\pm I \\leq\\mu|K+ia|,$\n\\item\n$K$ is bounded from below and $ I \\geq -\\mu|K+ia|.$\n\\end{compactenum}\nLet $H=K+I$ be the form sum of $K$ and $I$, so $H$ has as domain the\nset  of $u\\in\\mathcal{G}$ such that $Ku+Iu\\in\\mathcal{H}$ and acts as $Hu=Ku+Iu$. \nThen $H$ is a self-adjoint operator on $\\mathcal{H}$. If there is\n$\\alpha>1\/2$ such that \n$\\langle K\\rangle^{-1\/2}I\\langle K\\rangle^{-\\alpha}\\in\\mathscr{C}$ then $H$\nis strictly affiliated to $\\mathscr{C}$. \nIf $O\\in\\mathcal{S}$ and the semilattice $\\mathcal{S}$ is atomic then\n\\begin{equation}\\label{eq:ess2}\n\\mathrm{Sp_{ess}}(H)=\\overline{\\textstyle{\\bigcup}}_{E\\in\\mathcal{P}(\\mathcal{S})}\\mathrm{Sp}(H_{\\geq E}).\n\\end{equation}\n\\end{theorem}\n\nThe last assertion of the theorem follows immediately from Theorem\n\\ref{th:gas} and is a general version of the HVZ theorem.  In order\nto have a more explicit description of the observables $H_{\\geq\n  E}\\equiv\\mathscr{P}_{\\geq E}H$ we now prove an analog of Theorem 3.5 from\n\\cite{DG3}. We cannot use that theorem in our context for three\nreasons: first we did not suppose that $\\mathcal{S}$ has a maximal element,\nthen even if $\\mathcal{S}$ has a maximal element $\\mathcal{X}$ the action of the\ncorresponding algebra $\\mathscr{C}(\\mathcal{X})$ on the algebras $\\mathscr{C}(E)$ is\ndegenerate, and finally our ``free'' operator $K$ is not affiliated\nto $\\mathscr{C}(\\mathcal{X})$.\n\n\\begin{theorem}\\label{th:afi}\nFor each $E\\in\\mathcal{S}$ let $I(E)\\in L(\\mathcal{G},\\mathcal{G}^*)$ be a symmetric\noperator such that:\n\\begin{compactenum}[(i)]\n\\item\n$\\jap{K}^{-1\/2}I(E)\\jap{K}^{-\\alpha}\\in\\mathscr{C}(E)$ for some\n$\\alpha> 1\/2$ independent of $E$,\n\\item\nthere are real positive numbers $\\mu_E,a$ such that either $\\pm\nI(E) \\leq\\mu_E|K+ia|$ for all $E$ or $K$ is bounded from below and\n$ I(E) \\geq -\\mu_E|K+ia|$ for all $E$,\n\\item\nwe have $\\sum_E\\mu_E\\equiv\\mu<1$ and the series $\\sum_E I(E)\\equiv I$\nis norm summable in $L(\\mathcal{G},\\mathcal{G}^*)$.\n\\end{compactenum}\nLet us set $I_{\\geq E}=\\sum_{F\\geq E}I(F)$. Define the self-adjoint\noperator $H=K+I$ on $\\mathcal{H}$ as in Theorem \\ref{th:af} and define\nsimilarly the self-adjoint operator $H_{\\geq E}=K_{\\geq E}+I_{\\geq\n  E}$ on $\\mathcal{H}_{\\geq E}$.  Then the operator $H$ is strictly\naffiliated to $\\mathscr{C}$, the operator $H_{\\geq E}$ is strictly\naffiliated to $\\mathscr{C}_{\\geq E}$, and we have $\\mathscr{P}_{\\geq E}H=H_{\\geq E}$.\n\\end{theorem}\n\\proof  We shall consider only the case when $\\pm I(E)\n\\leq\\mu_E|K+ia|$ for all $E$. The more singular situation when $K$\nis bounded from below but there is no restriction on the positive\npart of the operators $I(E)$ (besides summability) is more difficult\nbut the main idea has been explained in \\cite{DG3}.\n\nWe first make some comments to clarify the definition of the\noperators $H$ and $H_{\\geq E}$.  Observe that our assumptions imply\n$\\pm I\\leq\\mu|K+ia|$ hence if we set\n\\[\n\\Lambda\\equiv|K+ia|^{-1\/2}=(K^2+a^2)^{-1\/4}\\in \\mathscr{T}_\\text{d}\n\\]\nthen  we obtain\n\\[\n\\pm\\braket{u}{Iu}\\leq\\mu\\braket{u}{|K+ia|u}=\n\\mu\\| |K+ia|^{1\/2}u\\| = \\mu\\| \\Lambda^{-1} u\\|\n\\] \nwhich is equivalent to $\\pm\\Lambda I\\Lambda\\leq\\mu$ or $\\|\\Lambda\nI\\Lambda\\|\\leq\\mu$. In particular we may use Theorem \\ref{th:af}\nin order to define the self-adjoint operator $H$. Moreover, we have\n\\[\n\\jap{K}^{-1\/2}I\\jap{K}^{-\\alpha}={\\textstyle\\sum_E}\n\\jap{K}^{-1\/2}I(E)\\jap{K}^{-\\alpha}\\in\\mathscr{C}\n\\]\nbecause the series is norm summable in $L(\\mathcal{H})$. Thus $H$ is\nstrictly affiliated to $\\mathscr{C}$.  \n\nIn order to define $H_{\\geq E}$ we first make a remark on \n$I_{\\geq E}$.  If we set $\\mathcal{G}_X=D(|K_X|^{-1\/2})$ and if we equip\n$\\mathcal{G}$ and $\\mathcal{G}_X$ with the norms  \\label{p:formd}\n\\[\n\\|u\\|_\\mathcal{G}=\\|\\jap{K}^{1\/2}u\\|_\\mathcal{H} \\quad \\text{and} \\quad\n\\|u\\|_{\\mathcal{G}_X}=\\|\\jap{K_X}^{1\/2}u\\|_{\\mathcal{H}_X}\n\\]\nrespectively then clearly $\\mathcal{G}=\\oplus_X\\mathcal{G}_X$ and\n$\\mathcal{G}^*=\\oplus_X\\mathcal{G}^*_X $ where the sums are Hilbertian direct sums\nand $\\mathcal{G}^*$ and $\\mathcal{G}_X^*$ are equipped with the dual norms.  Then\neach $I(F)$ may be represented as a matrix\n$I(F)=(I_{XY}(F))_{X,Y\\in\\mathcal{S}}$ of continuous operators\n$I_{XY}(E):\\mathcal{G}_Y\\to\\mathcal{G}^*_X$. Clearly\n\\[\n\\jap{K}^{-1\/2}I(F)\\jap{K}^{-\\alpha}=\n\\left(\\jap{K_X}^{-1\/2}I_{XY}(F)\\jap{K_Y}^{-\\alpha}\\right)_{X,Y\\in\\mathcal{S}}\n\\] \nand since by assumption (i) this belongs to $\\mathscr{C}(F)$ we see that\n$I_{XY}(F)=0$ if $X\\not\\supset F$ or $Y\\not\\supset F$. Now fix $E$\nand let $F\\supset E$. Then, when viewed as a sesquilinear form,\n$I(F)$ is supported by the subspace $\\mathcal{H}_{\\geq E}$ and has domain\n$\\mathcal{G}_{\\geq E}= D(|K_{\\geq E}|^{1\/2}$.  It follows that $I_{\\geq E}$ \nis a sesquilinear form with domain $\\mathcal{G}_{\\geq E}$ supported by the\nsubspace $\\mathcal{H}_{\\geq E}$ and may be thought as an element of\n$L(\\mathcal{G}_{\\geq E},\\mathcal{G}^*_{\\geq E})$ such that\n$\\pm I_{\\geq E}\\leq \\mu |K_{\\geq E}+ia|$ because \n$\\sum_{F\\supset E}\\mu_F\\leq \\mu$. To conclude, we may now define \n$H_{\\geq E}=K_{\\geq E}+I_{\\geq E}$ exactly as in the case of $H$ and\nget a self-adjoint operator on $\\mathcal{H}_{\\geq E}$ strictly affiliated to\n$\\mathscr{C}_{\\geq E}$. Note that this argument also gives\n\\begin{equation}\\label{eq:ek}\n\\jap{K}^{-1\/2} I(F) \\jap{K}^{-1\/2}=\n\\jap{K_{\\geq E}}^{-1\/2} I(F) \\jap{K_{\\geq E}}^{-1\/2}.\n\\end{equation}\nIt remains to be shown that $\\mathscr{P}_{\\geq E}H=H_{\\geq E}$. If we set\n$R\\equiv(ia-H)^{-1}$ and $R_{\\geq E}\\equiv(ia-H_{\\geq E})^{-1}$ then\nthis is equivalent to $\\mathscr{P}_{\\geq E}R=R_{\\geq E}$.  Let us set\n\\[\nU=|ia-K|(ia-K)^{-1}=\\Lambda^{-2}(ia-K)^{-1}, \\quad\nJ=\\Lambda I\\Lambda U.\n\\]\nThen $U$ is a unitary operator and $\\|J\\|<1$, so we get a norm\nconvergent series expansion\n\\[\nR=(ia-K-I)^{-1}=\n\\Lambda U(1-\\Lambda I\\Lambda U)^{-1}\\Lambda =\n{\\textstyle\\sum_{n\\geq0}}\\Lambda U J^n\\Lambda\n\\]\nwhich implies\n\\[\n\\mathscr{P}_{\\geq E} (R)=\n{\\textstyle\\sum_{n\\geq0}}\n\\mathscr{P}_{\\geq E}\\big(\\Lambda U J^{n}\\Lambda\\big)\n\\]\nthe series being norm convergent. Thus it suffices to prove that for\neach $n\\geq0$ \n\\begin{equation}\\label{eq:ekk}\n\\mathscr{P}_{\\geq E}\\big(\\Lambda U J^{n}\\Lambda\\big)=\n\\Lambda_{\\geq E} (J_{\\geq E})^{n}\\Lambda_{\\geq E}\n\\end{equation}\nwhere $J_{\\geq E}=\\Lambda_{\\geq E} I_{\\geq E}\\Lambda_{\\geq E}\nU_{\\geq E}$.  Here $\\Lambda_{\\geq E}$ and $U_{\\geq E}$ are\nassociated to $K_{\\geq E}$ in the same way $\\Lambda$ and $K$ are\nassociated to $K$. For $n=0$ this is obvious because \n$\\mathscr{P}_{\\geq E}K=K_{\\geq E}$. If $n=1$ this is easy because\n\\begin{align}\\label{eq:e}\n\\Lambda U J\\Lambda &= \\Lambda U \\Lambda I \\Lambda U\\Lambda=\n(ia-K)^{-1} I (ia-K)^{-1} \\\\\n&=\n[(ia-K)^{-1}\\jap{K}^{1\/2}] \\cdot \n[\\jap{K}^{-1\/2} I \\jap{K}^{-\\alpha}] \\cdot\n[\\jap{K}^{\\alpha} (ia-K)^{-1}] \\nonumber\n\\end{align}\nand it suffices to note that \n$\\mathscr{P}_{\\geq E}(\\jap{K}^{-1\/2} I(F) \\jap{K}^{-\\alpha})=0$ if\n$F\\not\\supset E$ and to use \\eqref{eq:ek} for $F\\supset E$. \n\nTo treat the general case we make some preliminary remarks. \nIf $J(F)=\\Lambda I(F) \\Lambda U$ then $J=\\sum_F J(F)$ where the\nconvergence holds in norm on $\\mathcal{H}$ because of the condition (iii). \nThen we have a norm convergent expansion\n\\[\n\\Lambda U J^n \\Lambda ={\\textstyle\\sum_{F_1,\\dots,F_n\\in\\mathcal{S}}}\n\\Lambda U J(F_1)\\dots J(F_n) \\Lambda.\n\\]\nAssume that we have shown $\\Lambda U J(F_1)\\dots\nJ(F_n)\\Lambda\\in\\mathscr{C}(F_1\\cap\\dots\\cap F_n)$.  Then we get\n\\begin{equation}\\label{eq:ekkk}\n\\mathscr{P}_{\\geq E}(\\Lambda U J^n \\Lambda)=\n{\\textstyle\\sum_{F_1\\geq E,\\dots,F_n\\geq E}}\n\\Lambda U J(F_1)\\dots J(F_n) \\Lambda\n\\end{equation}\nbecause if one $F_k$ does not contain $E$ then the intersection\n$F_1\\cap\\dots\\cap F_n$ does not contain $E$ hence $\\mathscr{P}_{\\geq E}$\napplied to the corresponding term gives $0$. Because of\n\\eqref{eq:ek} we have $J(F)=\\Lambda_{\\geq E} I(F) \\Lambda_{\\geq E}\nU_{\\geq E}$ if $F\\supset E$ and we may replace everywhere in the\nright hand side of \\eqref{eq:ekkk} $\\Lambda$ and $U$ by\n$\\Lambda_{\\geq E}$ and $U_{\\geq E}$. This clearly proves\n\\eqref{eq:ekk}. \n\nNow we prove the stronger fact \n$\\Lambda U J(F_1)\\dots J(F_n)\\in\\mathscr{C}(F_1\\cap\\dots\\cap F_n)$.  \nIf $n=1$ this follows from a slight modification of\n\\eqref{eq:e}: the last factor on the right hand side of \\eqref{eq:e}\nis missing but is not needed. Assume that the assertion holds for\nsome   $n$. Since $K$ is strictly affiliated to $\\mathscr{T}_\\text{d}$ and\n$\\mathscr{T}_\\text{d}$ acts non-degenerately on each $\\mathscr{C}(F)$ we may use the\nCohen-Hewitt theorem to deduce that there is $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$\nsuch that\n$\\Lambda U J(F_1)\\dots J(F_n)=T\\varphi(K)$ \nfor some $T\\in\\mathscr{C}(F_1\\cap\\dots\\cap F_n)$. \nThen\n\\[\n\\Lambda U J(F_1)\\dots J(F_n)J(F_{n+1})=T\\varphi(K)J(F_{n+1})\n\\]\nhence it suffices to prove that $\\varphi(K)J(F)\\in\\mathscr{C}(F)$ for any\n$F\\in\\mathcal{S}$ and any $\\varphi\\in\\cc_{\\mathrm{o}}(\\mathbb{R})$. But the set of $\\varphi$\nwhich have this property is a closed subspace of $\\cc_{\\mathrm{o}}(\\mathbb{R})$ which\nclearly contains the functions $\\varphi(\\lambda)=(\\lambda -z)^{-1}$\nif $z$ is not real hence is equal to $\\cc_{\\mathrm{o}}(\\mathbb{R})$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\\begin{remark}\\label{re:alpha}\nChoosing $\\alpha>1\/2$ allows one to consider perturbations of $K$\nwhich are of the same order as $K$, e.g. in the $N$-body situations\none may add to the Laplacian $\\Delta$ on operator like $\\nabla^*\nM\\nabla$ where the function $M$ is bounded measurable and has the\nstructure of an $N$-body type potential, cf. \\cite{DG3,DerI}.  \n\\end{remark}\n\n\nThe only assumption of Theorem \\ref{th:afi} which is really relevant\nis $\\jap{K}^{-1\/2}I(E)\\jap{K}^{-\\alpha}\\in\\mathscr{C}(E)$.  We shall give\nbelow more explicit conditions which imply it.  If we change\nnotation $E\\to Z$ and use the formalism introduced in the proof of\nTheorem \\ref{th:afi} we have\n\\begin{equation}\\label{eq:xye}\nI(Z)=(I_{XY}(Z))_{X,Y\\in\\mathcal{S}} \\quad \\text{with} \\quad\nI_{XY}(Z):\\mathcal{G}_Y\\to\\mathcal{G}^*_X \\text{ continuous}.\n\\end{equation}\nWe are interested in conditions on $I_{XY}(Z)$ which imply\n\\begin{equation}\\label{eq:xye1}\n\\jap{K_X}^{-1\/2}I_{XY}(Z)\\jap{K_X}^{-\\alpha} \\in \\mathscr{C}_{XY}(Z).\n\\end{equation}\nFor this we shall use Theorem \\ref{th:yzintr} which gives a simple\nintrinsic characterization of $\\mathscr{C}_{XY}(Z)$.\n\nThe construction which follows is interesting only if $X$ is not a\ndiscrete group, otherwise $X^*$ is compact and many conditions are\ntrivially satisfied. We shall use weights only in order to avoid\nimposing on the functions $h_X$ regularity conditions stronger than\ncontinuity.\n\nA positive function $w$ on $X^*$ is a \\emph{weight} if\n$\\lim_{k\\to\\infty} w(k)=\\infty$ and $w(k+p)\\leq\\omega(k)w(p)$ for\nsome function $\\omega$ on $X^*$ and all $k,p$. We say that $w$ is\n\\emph{regular} if one may choose $\\omega$ such that\n$\\lim_{k\\to0}\\omega(k)=1$.  The example one should have in mind when\n$X$ is an Euclidean space is $w(k)=\\jap{k}^s$ for some $s>0$. Note\nthat we have $\\omega(-k)^{-1}\\leq w(k+p)w(p)^{-1} \\leq \\omega(k)$\nhence if $w$ is a regular weight then \\label{p:regw}\n\\begin{equation}\\label{eq:regw}\n\\theta(k)\\equiv \\sup_{p\\in X^*}\\frac{|w(k+p)-w(p)|}{w(p)} \n\\Longrightarrow\n\\lim_{k\\to0}\\theta(k)=0.\n\\end{equation}\nIt is clear that if $w$ is a regular weight and $\\sigma\\geq 0$ is a\nreal number then $w^\\sigma$ is also a regular weight.\n\nWe say that two functions $f,g$ defined on a neighborhood of \ninfinity of $X^*$ are \\emph{equivalent} and we write $f\\sim g$ if\nthere are numbers $a,b$ such that $a|f(k)|\\leq|g(k)|\\leq\nb|f(k)|$. Then $|f|^\\sigma\\sim|g|^\\sigma$ for all $\\sigma>0$.\n\n\nWe denote $\\mathcal{G}^\\sigma_X=D(|K_X|^{\\sigma\/2})$ and\n$\\mathcal{G}^{-\\sigma}_X\\equiv(\\mathcal{G}^{\\sigma}_X)^*$ with $\\sigma \\geq 1$.  In\nparticular $\\mathcal{G}^1_X=\\mathcal{G}_X$ and $\\mathcal{G}^{-1}_X=\\mathcal{G}^*_X$.\n\n\\begin{proposition}\\label{pr:tex}\n  Assume that $h_X,h_Y$ are equivalent to regular weights. Let\n  $Z\\subset X\\cap Y$ and let $I_{XY}(Z)$ be a continuous map\n  $\\mathcal{G}_Y\\to\\mathcal{G}^*_X$ such that\n\\begin{enumerate}\n\\item\n$U_z I_{XY}(Z)=I_{XY}(Z) U_z$ if $z\\in Z$ and\n$V^*_k I_{XY}(Z) V_k\\to I_{XY}(Z)$ if $k\\to 0$ in $(X+Y)^*$,\n\\item \n$(U_x-1)I_{XY}(Z)\\to 0$ if $x\\to 0$ in $X$ and\n$(V_k-1) I_{XY}(Z)\\to 0$ if $k\\to 0$ in $(X\/Z)^*$,\n\\end{enumerate}\nwhere the limits hold in norm in $L(\\mathcal{G}^{\\sigma}_Y,\\mathcal{G}^{-1}_X)$ for\nsome $\\sigma\\geq1$. Then \\eqref{eq:xye1} holds with\n$\\alpha=\\sigma\/2$.\n\\end{proposition}\n\\proof We begin with some general comments on weights. Let $w$ be a\nregular weight and let $\\mathcal{G}_X$ be the domain of the operator $w(P)$\nin $\\mathcal{H}_X$ equipped with the norm $\\|w(P)u\\|$. Then $\\mathcal{G}_X$ is a\nHilbert space and if $\\mathcal{G}^*_X$ is its adjoint space then we get a\nscale of Hilbert spaces $\\mathcal{G}_X\\subset\\mathcal{H}_X\\subset\\mathcal{G}^*_X$ with\ncontinuous and dense embeddings. Since $U_x$ commutes with $w(P)$ it\nis clear that $\\{U_x\\}_{x\\in X}$ induces strongly continuous unitary\nrepresentation of $X$ on $\\mathcal{G}_X$ and $\\mathcal{G}^*_X$.  Then\n\\[\n\\|V_k u\\|_{\\mathcal{G}_X}=\\|w(k+P)u\\|\\leq\\omega(k)\\|u\\|_{\\mathcal{G}_X}\n\\]\nfrom which it follows that $\\{V_k\\}_{k\\in X^*}$ induces by\nrestriction and extension strongly continuous representations of\n$X^*$ in $\\mathcal{G}_X$ and $\\mathcal{G}^*_X$. Moreover, as operators on $\\mathcal{H}_X$\nwe have \\label{p:RK}\n\\begin{align}  \n|V_k^*w(P)^{-1}V_k-w(P)^{-1}| \n&=|w(k+P)^{-1}-w(P)^{-1}| \n= |w(k+P)^{-1}(w(P)-w(k+P))w(P)^{-1}|  \\nonumber \\\\\n& \\leq \\omega(-k)|(w(P)-w(k+P))w(P)^{-2}|\n\\leq \\omega(-k)\\theta(k) w(P)^{-1}. \\label{eq:refw} \n\\end{align}\nNow let $w_X,w_Y$ be regular weights equivalent to\n$|h_X|^{1\/2},|h_Y|^{1\/2}$ and let us set $S=I_{XY}(Z)$. Then\n\\[\n\\jap{K_X}^{-1\/2}S\\jap{K_Y}^{-\\alpha}=\n\\jap{K_X}^{-1\/2}w_X(P)\\cdot \nw_X(P)S w_Y(P)^{-2\\alpha} \\cdot\nw_Y(P)^{2\\alpha}\\jap{K_Y}^{-\\alpha}\n\\]\nand $\\jap{h_X}^{-1\/2}w_X$, $\\jap{h_Y}^{-\\alpha}w_Y^{2\\alpha}$ and\ntheir inverses are bounded continuous functions on $X,Y$. Since\n$\\mathscr{C}_{XY}(Z)$ is a non-degenerate left $\\mathscr{T}_X$-module and right\n$\\mathscr{T}_Y$-module we may use the Cohen-Hewitt theorem to deduce that\n\\eqref{eq:xye1} is equivalent to\n\\begin{equation}\\label{eq:xye2}\nw_X(P)^{-1}I_{XY}(Z) w_Y(P)^{-\\sigma} \\in \\mathscr{C}_{XY}(Z)\n\\end{equation}\nwhere $\\sigma=2\\alpha$.  To simplify notations we set $W_X=w_X(P),\nW_Y=w^\\sigma_Y(P)$.  We also omit the index $X$ or $Y$ for the\noperators $W_X,W_Y$ since their value is obvious from the\ncontext. In order to show $W^{-1}SW^{-1}\\in \\mathscr{C}_{XY}(Z)$ we check\nthe conditions of Theorem \\ref{th:yzintr} with $T=W^{-1}SW^{-1}$.\nThe first part of condition (2) of the theorem is verified by the\nhypothesis (2) of the present proposition.  We may assume $\\sigma>1$\nand then hence the second part of condition (2) of the theorem\nfollows from\n\\[\n\\|T(U_y-1)\\|\\leq \\|W^{-1}I_{XY}(Z)w_Y^{-1}(P)\\|\\to 0 \n\\|(U_y-1)w_Y^{1-\\sigma}(P)\\| \\quad \\text{if } y\\to0.\n\\]\nTo check the second part of condition (1) of the theorem set\n$W_k=V_k^*WV_k$ and $S_k=V_k^* SV_k$ and write\n\\begin{align*}\nV_k^*TV_k-T \n&= W_k^{-1} S_k W_k^{-1}-W^{-1}SW^{-1}\\\\\n&= (W_k^{-1}-W^{-1})S_kW_k^{-1} + W^{-1}(S_k-S)W_k^{-1}\n+W^{-1}S(W_k^{-1} -W^{-1}).\n\\end{align*}\nNow if we use \\eqref{eq:refw} and set $\\xi(k)=\\omega(-k)\\theta(k)$\nwe get\n\\begin{align*}\n\\|V_k^*TV_k-T\\| &\\leq \n\\xi(k)\\|W^{-1}S_kW_k^{-1}\\| + \\|W^{-1}(S_k-S)W^{-1}\\|\\|W W_k^{-1}\\|\n+\\xi(k)\\|W^{-1}SW^{-1}\\|\n\\end{align*}\nwhich clearly tends to zero if $k\\to0$. Condition\n(3) of Theorem \\ref{th:yzintr} follows by a similar argument.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\nNow let $H$ be defined according to the algorithm of \\S\\ref{ss:ex}.\nThen condition (i) of Theorem \\ref{th:afi} will be satisfied for all\n$\\alpha >1\/2$. Indeed, from Proposition \\ref{pr:tex} we get\n$\\jap{K}^{-1\/2}\\Pi_\\mathcal{T} I(Z)\\Pi_\\mathcal{T}\\jap{K}^{-\\alpha}\\in\\mathscr{C}(Z)$ for\nany finite $\\mathcal{T}$ and this operator converges in norm to\n$\\jap{K}^{-1\/2} I(Z)\\jap{K}^{-\\alpha}$.  Thus all conditions of\nTheorem \\ref{th:afi} are fulfilled by the Hamiltonian $H=K+I$ and so\n$H$ is strictly affiliated to $\\mathscr{C}$.  \\label{p:algor}\n\n\n\n\\section{The Mourre estimate} \n\\label{s:mou}\n\\protect\\setcounter{equation}{0}\n\n\n\n\\subsection{Proof of the Mourre estimate}\n\\label{ss:mest}\n\nFrom now on we work in the framework of the second part of Section\n\\ref{s:euclid}, so we assume that $\\mathcal{S}$ is a \\emph{finite}\nsemilattice of finite dimensional subspaces of an Euclidean\nspace. In this subsection we prove the Mourre estimate for\nnonrelativistic Hamiltonians.  The strategy of the proof is that\nintroduced in \\cite{BG2} and further developed in \\cite{ABG,DG2}\n(graded $C^*$-algebras over infinite semilattices and dispersive\nHamiltonians are considered in Section 5 from \\cite{DG2}).  We\nchoose the generator $D$ of the dilation group $W_\\tau$ in $\\mathcal{H}$ as\nconjugate operator for reasons explained below. For special types of\ninteractions, similar to those occurring in quantum field models,\nwhich are allowed by our formalism, better choices can be made, but\nat a technical level there is nothing new in that with respect to\n\\cite{Geo} (these special interactions correspond to distributive\nsemilattices $\\mathcal{S}$).\n\n\n\n\nThe dilations implement a group of automorphisms of the\n$C^*$-algebra $\\mathscr{C}$ which is compatible with the grading, i.e. it\nleaves invariant each component $\\mathscr{C}(Z)$ of $\\mathscr{C}$.  In fact, it is\nclear that $W_\\tau^*\\mathscr{C}_{XY}(Z)W_\\tau=\\mathscr{C}_{XY}(Z)$ for all $X,Y,Z$\nhence $W_\\tau^*\\mathscr{C}(Z)W_\\tau=\\mathscr{C}(Z)$.  This fact plays a fundamental\nrole in the proof of the Mourre estimate for operators affiliated to\n$\\mathscr{C}$ and explains the choice of $D$ as conjugate operator.\nMoreover, for each $T\\in\\mathscr{C}$ the map $\\tau\\mapsto W^*_\\tau T W_\\tau$\nis norm continuous. We can compute explicitly the function\n$\\widehat\\rho_H$ thanks to the relation\n\\begin{equation}\\label{eq:dlap}\nW^*_\\tau \\Delta_X W_\\tau = \\mathrm{e}^\\tau\\Delta_X \\quad \\text{or}\\quad\n[\\Delta_X, i D]=\\Delta_X \n\\end{equation}\nWe say that a self-adjoint operator $H$ \\emph{is of class $C^1(D)$}\nor \\emph{of class $C^1_\\mathrm{u}(D)$} if $W^*_\\tau RW_\\tau$ as a function\nof $\\tau$ is of class $C^1$ strongly or in norm respectively.  Here\n$R=(H-z)^{-1}$ for some $z$ outside the spectrum of $H$.  The formal\nrelation\n\\begin{equation}\\label{eq:dres}\n[D,R]= R[H,D] R \n\\end{equation}\ncan be given a rigorous meaning as follows. If $H$ is of class\n$C^1(D)$ then the intersection $\\mathscr{D}$ of the domains of the operators\n$H$ and $D$ is dense in $D(H)$ and the sesquilinear form with domain\n$\\mathscr{D}$ associated to the formal expression $HD-DH$ is continuous for\nthe topology of $D(H)$ so extends uniquely to a continuous\nsesquilinear form on the domain of $H$ which is denoted\n$[H,D]$. This defines the right hand side of \\eqref{eq:dres}.  The\nleft hand side can be defined for example as \n$i\\frac{d}{d\\tau}W_\\tau^*RW_\\tau|_{\\tau=0}$. \n\nFor Hamiltonians as those considered here it is easy to decide that\n$H$ is of class $C^1(D)$ in terms of properties of the commutator\n$[H, D]$.  Moreover, the following is easy to prove: \\emph{if $H$\n  is affiliated to $\\mathscr{C}$ then $H$ is of class $C^1_\\mathrm{u}(D)$ if and\n  only if $H$ is of class $C^1(D)$ and $[R,D]\\in\\mathscr{C}$}.\n\nLet $H$ be of class $C^1(D)$ and $\\lambda\\in\\mathbb{R}$. Then for each\n$\\theta\\in\\cc_{\\mathrm{c}}(\\mathbb{R})$ with $\\theta(\\lambda)\\neq0$ one may find a real\nnumber $a$ and a compact operator $K$ such that \n\\begin{equation}\\label{eq:must}\n\\theta(H)^*[H,iD]\\theta(H)\\geq a|\\theta(H)|^2+K.\n\\end{equation}\n\n\\begin{definition}\\label{df:must}\nThe upper bound $\\widehat\\rho_H(\\lambda)$ of the numbers $a$ for which\nsuch an estimate holds is \\emph{the best constant in the Mourre\n  estimate for $H$ at $\\lambda$}.  The \\emph{threshold set} of $H$\n(relative to $D$) is the closed real set\n\\begin{equation}\\label{eq:thr0}\n\\tau(H)=\\{\\lambda \\mid \\widehat\\rho_H(\\lambda)\\leq0\\}\n\\end{equation}\nOne says that $D$ is \\emph{conjugate to} $H$ at $\\lambda$ if \n$\\widehat\\rho_H(\\lambda)>0$. \n\\end{definition}\n\nThe set $\\tau(H)$ is closed because the function\n$\\widehat\\rho_H:\\mathbb{R}\\to]-\\infty,\\infty]$ is lower semicontinuous.\n\n\nTo each closed real set $A$ we associate the function\n$N_A:\\mathbb{R}\\to[-\\infty,\\infty[$ defined by\n\\begin{equation}\\label{eq:na}\nN_A(\\lambda)=\\sup\\{ x\\in A \\mid x\\leq\\lambda\\}.\n\\end{equation}\nWe make the convention $\\sup\\emptyset=-\\infty$. Thus $N_A$ may take\nthe value $-\\infty$ if and only if $A$ is bounded from below and then\n$N_A(\\lambda)=-\\infty$ if and only if $\\lambda<\\min A$. The function\n$N_A$ is further discussed during the proof of Lemma \\ref{lm:nab}.\n\n\nNonrelativistic many-body Hamiltonians have been introduced in\nDefinition \\ref{df:NR}.  Let $\\mathrm{ev}(T)$ be the set of\neigenvalues of an operator $T$.\n\n\\begin{theorem}\\label{th:thr}\n  Let $\\mathcal{S}$ be finite and let $H=H_\\mathcal{S}$ be a nonrelativistic\n  many-body Hamiltonian of class $C^1_\\mathrm{u}(D)$. Then\n  $\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ for all real\n  $\\lambda$ and $\\tau(H)$ is a closed \\emph{countable} real set\n  given by\n\\begin{equation}\\label{eq:thr}\n\\tau(H)=\\textstyle{\\bigcup}_{X\\neq O}\\mathrm{ev}(H_{\\mathcal{S}\/X}).\n\\end{equation} \n\\end{theorem}\n\\proof We first treat the case $O\\in\\mathcal{S}$. We need some facts which\nare discussed in detail in Sections 7.2, 8.3 and 8.4 from \\cite{ABG}\n(see pages 51--61 in \\cite{BG2} for a shorter presentation).\n\n(i) For each real $\\lambda$ let $\\rho_H(\\lambda)$ be the upper bound\nof the numbers $a$ for which an estimate like \\eqref{eq:must} but\nwith $K=0$ holds. This defines a lower semicontinuous function\n$\\rho_H:\\mathbb{R}\\to ]-\\infty,\\infty] $ hence the set\n$\\varkappa(H)=\\{\\lambda \\mid \\rho_H(\\lambda)\\leq 0\\}$ is a closed\nreal set called \\emph{critical set} of $H$ (relative to $D$). We\nclearly have $\\rho_H\\leq\\widehat\\rho_H$ and so\n$\\tau(H)\\subset\\varkappa(H)$.\n\n(ii) Let $\\mu(H)$ be the set of eigenvalues of $H$ such that\n$\\widehat\\rho_H(\\lambda)>0$. Then $\\mu(H)$ is a discrete subset of\n$\\mathrm{ev}(H)$ consisting of eigenvalues of finite\nmultiplicity. This is essentially the virial theorem.\n\n(iii) There is a simple and rather unexpected relation between the\nfunctions $\\rho_H$ and $\\widehat\\rho_H$: they are ``almost'' equal. In\nfact, $\\rho_H(\\lambda)=0$ if $\\lambda\\in\\mu(H)$ and\n$\\rho_H(\\lambda)=\\widehat\\rho_H(\\lambda)$ otherwise. In particular\n\\begin{equation}\\label{eq:tev}\n\\varkappa(H)=\\tau(H)\\cup \\mathrm{ev}(H)=\\tau(H)\\sqcup\\mu(H)\n\\end{equation}\nwhere $\\sqcup$ denotes disjoint union. \n\n(iv) This step is easy but rather abstract and the $C^*$-algebra\nsetting really comes into play. We assume that $H$ is affiliated to\nour algebra $\\mathscr{C}$. The preceding arguments did not require more than\nthe $C^1(D)$ class. Now we require $H$ to be of class $C^1_\\mathrm{u}(D)$.\nThen the operators $H_{\\geq X}$ are also of class $C^1_\\mathrm{u}(D)$ and\nwe have the important relation (Theorem 8.4.3 in \\cite{ABG} or\nTheorem 4.4 in \\cite{BG2})\n\\[\n\\widehat\\rho_H=\\min_{X\\in\\mathcal{P}(\\mathcal{S})}\\rho_{H_{\\geq X}}.\n\\]\nTo simplify notations we adopt the abbreviations $\\rho_{H_{\\geq\n    X}}=\\rho_{\\geq X}$ and instead of $X\\in\\mathcal{P}(\\mathcal{S})$ we write\n$X\\gtrdot O$, which should be read ``$X$ covers $O$''. For coherence\nwith later notations we also set $\\widehat\\rho_H=\\widehat\\rho_\\mathcal{S}$.  So\n\\eqref{eq:mustq} may be written\n\\begin{equation}\\label{eq:mustq}\n\\widehat\\rho_\\mathcal{S}=\\min_{X\\gtrdot O}\\rho_{\\geq X}.\n\\end{equation}\n\n(v) From \\eqref{eq:dlap} and \\eqref{eq:NR} we get\n\\[\nH_{\\geq X}=\\Delta_X\\otimes1+ 1\\otimes H_{\\mathcal{S}\/X}, \\quad\n[H_{\\geq X},iD]=\\Delta_X\\otimes1+ 1\\otimes [D,i H_{\\mathcal{S}\/X}].\n\\]\nRecall that we denote $D$ the generator of the dilation group\nindependently of the space in which it acts. We note that the formal\nargument which gives the second relation above can easily be made\nrigorous but this does not matter here. Indeed, since $H_{\\geq X}$\nis of class $C^1_\\mathrm{u}(D)$ and by using the first relation above, one\ncan easily show that $H_{\\mathcal{S}\/X}$ is also of class $C^1_\\mathrm{u}(D)$ (see\nthe proof of Lemma 9.4.3 in \\cite{ABG}). Now we may use Theorem\n8.3.6 from \\cite{ABG} to get\n\\[\n\\rho_{\\geq X}(\\lambda)=\\inf_{\\lambda_1+\\lambda_2=\\lambda}\n\\big(\\rho_{\\Delta_X}(\\lambda_1) + \\rho_{\\mathcal{S}\/X}(\\lambda_2) \\big)\n\\]\nwhere $\\rho_{\\mathcal{S}\/X}=\\rho_{H_{\\mathcal{S}\/X}}$. But clearly if $X\\neq O$ we\nhave $\\rho_{\\Delta_X}(\\lambda)=\\infty$ if $\\lambda<0$ and \n$\\rho_{\\Delta_X}(\\lambda)=\\lambda$ if $\\lambda\\geq0$. Thus we get\n\\begin{equation}\\label{eq:mustt}\n\\rho_{\\geq X}(\\lambda)=\\inf_{\\mu\\leq\\lambda}\n\\big(\\lambda-\\mu + \\rho_{\\mathcal{S}\/X}(\\mu) \\big)\n=\\lambda-  \\sup_{\\mu\\leq\\lambda}\\big(\\mu - \\rho_{\\mathcal{S}\/X}(\\mu) \\big).\n\\end{equation}\n\n(vi) Now from \\eqref{eq:mustq} and \\eqref{eq:mustt} we get\n\\begin{equation}\\label{eq:musr}\n\\lambda-\\widehat\\rho_\\mathcal{S}(\\lambda)=\n\\max_{X\\gtrdot O}\\sup_{\\mu\\leq\\lambda}\n\\big(\\mu - \\rho_{\\mathcal{S}\/X}(\\mu)\\big).\n\\end{equation}\nFinally, we are able to prove the formula\n$\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ by induction\nover the semilattice $\\mathcal{S}$. In other terms, we assume that the\nformula is correct if $H$ is replaced by $H_{\\mathcal{S}\/X}$ for all $X\\neq\nO$ and we prove it for $H=H_{\\mathcal{S}\/O}$. So we have to show that the\nright hand side of \\eqref{eq:musr} is equal to\n$N_{\\tau(H)}(\\lambda)$.\n\nAccording to step (iii) above we have $\\rho_{\\mathcal{S}\/X}(\\mu)=0$ if\n$\\mu\\in\\mu(H_{\\mathcal{S}\/X})$ and\n$\\rho_{\\mathcal{S}\/X}(\\mu)=\\widehat\\rho_{\\mathcal{S}\/X}(\\mu)$ otherwise. Since by the\nexplicit expression of $\\widehat\\rho_{\\mathcal{S}\/X}$ this is a positive\nfunction and since $\\rho_H(\\lambda)\\leq0$ is always true if\n$\\lambda$ is an eigenvalue, we get $\\mu-\\rho_{\\mathcal{S}\/X}(\\mu)=\\mu$ if\n$\\mu\\in\\mathrm{ev}(H_{\\mathcal{S}\/X})$ and\n\\[\n\\mu-\\rho_{\\mathcal{S}\/X}(\\mu)=\\mu-\\widehat\\rho_{\\mathcal{S}\/X}(\\mu)=\nN_{\\tau(H_{\\mathcal{S}\/X})}(\\mu)\n\\]\notherwise. From the first part of Lemma \\ref{lm:nab} below we get\n\\[\n\\sup_{\\mu\\leq\\lambda}\\big(\\mu - \\rho_{\\mathcal{S}\/X}(\\mu)\\big)=\nN_{\\mathrm{ev}(H_{\\mathcal{S}\/X}) \\cup \\tau(H_{\\mathcal{S}\/X})}.\n\\]\nIf we use the second part of Lemma \\ref{lm:nab} then we see that\n\\[\n\\max_{X\\gtrdot O}\\sup_{\\mu\\leq\\lambda}\\big(\\mu -\n\\rho_{\\mathcal{S}\/X}(\\mu)\\big)=\n\\max_{X\\gtrdot O}N_{\\mathrm{ev}(H_{\\mathcal{S}\/X}) \\cup \\tau(H_{\\mathcal{S}\/X})}\n\\]\nis the $N$ function of the set \n\\[\n\\bigcup_{X\\gtrdot O}\\big(\\mathrm{ev}(H_{\\mathcal{S}\/X}) \n\\cup \\tau(H_{\\mathcal{S}\/X})\\big)=\n\\bigcup_{X\\gtrdot O}\\left(\\mathrm{ev}(H_{\\mathcal{S}\/X}) \n\\bigcup \\bigcup_{Y>X}\\mathrm{ev}(H_{\\mathcal{S}\/Y})\\right)=\n\\bigcup_{X>O}\\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\]\nwhich finishes the proof of\n$\\widehat\\rho_H(\\lambda)=\\lambda-N_{\\tau(H)}(\\lambda)$ hence the proof\nof Theorem \\ref{th:thr} in the case $O\\in\\mathcal{S}$.\n\nNo assume $O\\notin\\mathcal{S}$ and let $E=\\min\\mathcal{S}$. Then $O\\in\\mathcal{S}\/E$ so we may\nuse the preceding result for $H_{\\mathcal{S}\/E}$. Moreover, we have\n$H=\\Delta_E\\otimes 1 + 1\\otimes H_{\\mathcal{S}\/E}$. Thus\n$\\mathrm{ev}(H)=\\emptyset$, $\\widehat\\rho_H=\\rho_H$, and we may use a\nrelation similar to \\eqref{eq:mustt} to get\n\\[\n\\lambda-\\widehat\\rho_H(\\lambda)=\n\\sup_{\\mu\\leq\\lambda}(\\mu-\\rho_{\\mathcal{S}\/E}(\\mu)).\n\\]\nBy what we have shown before we have\n$\\mu-\\rho_{\\mathcal{S}\/E}(\\mu)=N_{\\tau(H_{\\mathcal{S}\/E})}(\\mu)$ if\n$\\mu\\notin\\mu(H_{\\mathcal{S}\/E})$ and otherwise $\\mu-\\rho_{\\mathcal{S}\/E}(\\mu)=\\mu$.\nFrom Lemma \\ref{lm:nab} we get\n$\\lambda-\\widehat\\rho_H(\\lambda)=N_{\\tau(H_{\\mathcal{S}\/E})\\cup\\mu(H_{\\mathcal{S}\/E})}$.\nBut from \\eqref{eq:tev} we get $\\tau(H_{\\mathcal{S}\/E})\\cup\\mu(H_{\\mathcal{S}\/E})=\n\\tau(H_{\\mathcal{S}\/E})\\cup\\mathrm{ev}(H_{\\mathcal{S}\/E})$. From\n\\eqref{eq:thr} we get\n\\[\n\\tau(H_{\\mathcal{S}\/E})=\\textstyle{\\bigcup}_{Y\\in\\mathcal{S}\/E, Y\\neq O}\\mathrm{ev}(H_{(\\mathcal{S}\/E)\/Y})\n=\\textstyle{\\bigcup}_{X\\in\\mathcal{S}, X\\neq E} \\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\] \nbecause if we write $Y=X\/E$ with $X\\in\\mathcal{S}, X\\neq E$ then\n$(\\mathcal{S}\/E)\/(X\/E)=\\mathcal{S}\/X$. Finally,\n\\[\n\\tau(H_{\\mathcal{S}\/E})\\,\\textstyle{\\bigcup}\\,\\mathrm{ev}(H_{\\mathcal{S}\/E})=\n\\textstyle{\\bigcup}_{X\\in\\mathcal{S}} \\mathrm{ev}(H_{\\mathcal{S}\/X})\n\\]\nwhich proves the Theorem in the case $O\\notin\\mathcal{S}$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\nIt remains to show the following fact which was used above.\n\n\\begin{lemma}\\label{lm:nab}\nIf $A$ and $A\\cup B$ are closed and if $M$ is the function given by\n$M(\\mu)=N_A(\\mu)$ for $\\mu\\notin B$ and $M(\\mu)=\\mu$ for $\\mu\\in B$\nthen $\\sup_{\\mu\\leq\\lambda}M(\\mu)=N_{A\\cup B}(\\lambda)$.  If $A,B$\nare closed then $\\sup(N_A,N_B)=N_{A\\cup B}$.\n\\end{lemma}\n\\proof The last assertion of the lemma is easy to check, we prove\nthe first one.  Observe first that the function $N_A$ has the\nfollowing properties:\n\\begin{compactenum}\n\\item[(i)]\n$N_A$ is increasing and right-continuous,\n\\item[(ii)]\n$N_A(\\lambda)=\\lambda$ if $\\lambda\\in A$,\n\\item[(iii)] $N_A$ is locally constant and $N_A(\\lambda)<\\lambda$ on\n  $A^\\mathrm{c}\\equiv \\mathbb{R}\\setminus A$.\n\\end{compactenum} \nIndeed, the first assertion in (i) and assertion (ii) are obvious.\nThe second part of (i) follows from the more precise and easy to prove\nfact \n\\begin{equation}\\label{eq:nae}\nN_A(\\lambda+\\varepsilon)\\leq N_A(\\lambda)+\\varepsilon \\quad\n\\text{for all real } \\lambda \\text{ and } \\varepsilon>0.\n\\end{equation}\nA connected component of the open set $A^\\mathrm{c}$ is necessarily an\nopen interval of one of the forms $]-\\infty,y[$ or $]x,y[$ or\n$]x,\\infty[$ with $x,y\\in A$. On the first interval (if such an\ninterval appears) $N_A$ is equal to $-\\infty$ and on the second or\nthe third one it is clearly constant and equal to $N_A(x)$. We also\nnote that the function $N_A$ is characterized by the properties\n(i)--(iii).\n\nThus, if we denote $N(\\lambda)=\\sup_{\\mu\\leq\\lambda}M(\\mu)$, then it\nwill suffices to show that the function $N$ satisfies the conditions\n(i)--(iii) with $A$ replace by $A\\cup B$. Observe that $M(\\mu)\\leq\\mu$\nand the equality holds if and only if $\\mu\\in A\\cup B$.  Thus $N$ is\nincreasing, $N(\\lambda)\\leq\\lambda$, and $N(\\lambda)=\\lambda$ if\n$\\lambda\\in A\\cup B$.\n\nNow assume that $\\lambda$ belongs to a bounded connected component\n$]x,y[ $ of $A\\cup B$ (the unbounded case is easier to treat). If\n$x<\\mu<y$ then $\\mu\\notin B$ so $M(\\mu)=N_A(\\mu)$ and $]x,y[$ is\nincluded in a connected component of $A^\\mathrm{c}$ hence $M(\\mu)=N_A(x)$.\nThen $N(\\lambda)= \\max(\\sup_{\\nu\\leq x}M(\\nu),N_A(x))$ hence $N$ is \nconstant on $]x,y[$. Here we have $M(\\nu)\\leq\\nu\\leq x$ so if $x\\in A$\nthen $N_A(x)=x$ and we get $N(\\lambda)=x$. If $x\\in B\\setminus A$ then\n$M(x)=x$ so $\\sup_{\\nu\\leq x}M(\\nu)=x$ and $N_A(x)<x$ hence \n$N(\\lambda)=x$. Since $x\\in A\\cup B$ one of these two cases is \ncertainly realized and the same argument gives $N(x)=x$. Thus the \nvalue of $N$ on $]x,y[$ is $N(x)$ so $N$ is right continuous on \n$[x,y[$.  Thus we proved that $N$ is locally constant and right \ncontinuous on the complement of $A\\cup B$ and also that \n$N(\\lambda)<\\lambda$ there.\n\nIt remains to be shown that $N$ is right continuous at each point of\n$\\lambda\\in A\\cup B$. We show that \\eqref{eq:nae} hold with $N_A$\nreplaced by $N$. If $\\mu\\leq\\lambda$ then \n$M(\\mu)\\leq\\mu\\leq\\lambda=M(\\lambda)$ hence we have \n\\[\nN(\\lambda+\\varepsilon)=\n\\sup_{\\lambda\\leq\\mu\\leq\\lambda+\\varepsilon}M(\\mu). \n\\]\nBut $M(\\mu)$ above is either $N_A(\\mu)$ either $\\mu$. In the second\ncase $\\mu\\leq\\lambda+\\varepsilon$ and  in the first case\n\\[\nN_A(\\mu)\\leq N_A(\\lambda+\\varepsilon)\\leq N_A(\\lambda)+\\varepsilon\n\\leq \\lambda +\\varepsilon. \n\\]\nThus we certainly have $N(\\lambda+\\varepsilon)\\leq\\lambda+\\varepsilon$\nand $\\lambda=N(\\lambda)$ because $\\lambda\\in A\\cup B$.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\n\n\\subsection{A general class of interactions}\n\\label{ss:euc}\n\n\nThe rest of this section is devoted to some technical questions. Our\nmain purpose is to clarify the structure of the interactions in the\nEuclidean case.\n\nThe following compactness criterion will be useful. This is a\nconsequence of the Riesz-Kolmogorov theorem and of the argument page\n\\pageref{p:RK} involving the regularity of the weight. Let $E,F$ be\narbitrary Euclidean space and $s,t\\in\\mathbb{R}$.\n\n\\begin{proposition}\\label{pr:stsob}\n  An operator $T\\in L(\\mathcal{H}^s_E,\\mathcal{H}^t_F)$ is compact if and only if\n  one of the next two equivalent conditions is satisfied,\n  where $\\|\\cdot\\|$ is the norm in $L(\\mathcal{H}^s_E,\\mathcal{H}^t_F)$:\n\\begin{compactenum}\n\\item[{\\rm(i)}]\n$\\|(U_x-1)T\\| + \\|(V_x-1)T\\| \\to 0 \\quad \n\\text{if } x\\to 0 \\text{ in }F$,\n\\item[{\\rm(ii)}]\n$\\|T(U_x-1)\\| + \\|T(V_x-1)\\| \\to 0 \\quad \n\\text{if } x\\to 0 \\text{ in }E$.\n\\end{compactenum}\n\\end{proposition}\n\nWe denote $L^\\circ(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$ the set of small at infinity\noperators, cf.  Definition \\ref{df:small}. Clearly\n\\emph{$L^\\circ(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$ is a closed subspace of\n  $L(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$}.\n\n\\begin{corollary}\\label{co:small}\n  An operator $T\\in L(\\mathcal{H}^s_E,\\mathcal{H}^{t}_F)$ is small at infinity if\n  and only if $\\lim_{k\\to0}T(V_k-1)= 0$ in norm in\n  $L(\\mathcal{H}^{s+\\varepsilon}_E,\\mathcal{H}^{t}_F)$ for some $\\varepsilon>0$.\n  Then this holds for all $\\varepsilon>0$.\n\\end{corollary}\n\nIndeed, the first part of condition (ii) of Proposition\n\\ref{pr:stsob} ($s$ replaced by $s+\\varepsilon$) is\nautomatically satisfied.\n\nWe now give a Sobolev space version of Proposition \\ref{pr:tex}\nwhich uses the weights $\\jap{\\cdot}^s$ and is convenient in\napplications.  By using Theorem \\ref{th:xyzeintr} instead of Theorem\n\\ref{th:yzintr} in the proof of Proposition \\ref{pr:tex} we get:\n\n\\begin{proposition}\\label{pr:etex}\nLet $s,t>0$ and $Z\\subset X\\cap Y$. Let \n$I_{XY}(Z)\\in  L(\\mathcal{H}^t_Y,\\mathcal{H}^{-s}_X)$\nsuch that the following relations hold in norm in\n$ L(\\mathcal{H}^{t+\\varepsilon}_Y,\\mathcal{H}^{-s}_X)$ for some $\\varepsilon > 0$:\n\\begin{enumerate}\n\\item[{\\rm(1)}]\n$U_z I_{XY}(Z)=I_{XY}(Z) U_z$ if $z\\in Z$ and\n$V^*_z I_{XY}(Z) V_z\\to I_{XY}(Z)$ if $z\\to 0$ in $Z$,\n\\item[{\\rm(2)}] \n$I_{XY}(Z)(V_y-1)\\to 0$ if $y\\to 0$ in $Y\/Z$.\n\\end{enumerate}\nIf $h_X,h_Y$ are continuous real functions on $X,Y$ such that\n$h_X(x)\\sim\\jap{x}^{2s}$ and $h_Y(y)\\sim\\jap{y}^{2t}$ and if we set\n$K_X=h_X(P), K_Y=h_Y(P)$ then\n$\\jap{K_X}^{-1\/2}I_{XY}(Z)\\jap{K_Y}^{-\\alpha}\\in\\mathscr{C}_{XY}(Z)$ if\n$\\alpha>1\/2$.\n\\end{proposition}\n\nOur next purpose is to discuss in more detail the structure of the\noperators $I_{XY}(Z)$ from Proposition \\ref{pr:etex}. For this we\nmake a Fourier transformation $\\mathcal{F}_Z$ in the $Z$ variable as in the\nproof of Theorem \\ref{th:xyzeintr}.\n\nWe fix $X,Y,Z$ with $Z\\subset X\\cap Y$, use the tensor\nfactorizations \\eqref{eq:xyzint} and make identifications like\n$\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}= L^2(Z;\\mathcal{H}_{X\/Z})$. Thus\n$\\mathcal{H}_X=\\mathcal{H}_Z\\otimes\\mathcal{H}_{X\/Z}$ and $\\Delta_X=\\Delta_Z\\otimes 1 +\n1\\otimes \\Delta_{X\/Z}$ hence if $s\\geq0$\n\\begin{equation}\\label{eq:stens}\n\\mathcal{H}^s(X)=\\mathcal{H}(Z;\\mathcal{H}^s(X\/Z))\\cap \\mathcal{H}^s(Z;\\mathcal{H}_{X\/Z})=\n\\big(\\mathcal{H}_Z\\otimes\\mathcal{H}^s(X\/Z)\\big)\\cap \n\\big(\\mathcal{H}^s(Z)\\otimes\\mathcal{H}_{X\/Z}\\big)\n\\end{equation}\nwhere our notations are extended to  vector-valued Sobolev spaces. \nClearly\n\\begin{equation}\\label{eq:lap}\n\\mathcal{F}_Z \\jap{P_X}^s \\mathcal{F}_Z^{-1} = \n\\int_Z^\\oplus (1+|k|^2+|P_{X\/Z}|^2)^{s\/2} \\text{d} k.\n\\end{equation}\nThen from \\eqref{eq:CZ} and $\\mathscr{T}_Z=\\mathcal{F}_Z^{-1}\\cc_{\\mathrm{o}}(Z)\\mathcal{F}_Z$ we get\n\\begin{equation*}\n\\mathscr{C}_{XY}(Z)=\\mathscr{T}_Z\\otimes \\mathscr{K}_{X\/Z,Y\/Z}=\n\\mathcal{F}_Z^{-1}\\cc_{\\mathrm{o}}(Z;\\mathscr{K}_{X\/Z,Y\/Z})\\mathcal{F}_Z.\n\\end{equation*}\nTo each weakly measurable map $I_{XY}^Z:Z\\to\nL(\\mathcal{H}^t_{Y\/Z},\\mathcal{H}^{-s}_{X\/Z})$ such that\n\\begin{equation}\\label{eq:Iest}\n\\sup\\nolimits_k\n\\|(1+|k|+|P_{X\/Z}|)^{-s}I_{XY}^Z(k)(1+|k|+|P_{Y\/Z}|)^{-t}\\| <\\infty.\n\\end{equation} \nwe associate a continuous operator \n$I_{XY}(Z):\\mathcal{H}^t_Y\\to\\mathcal{H}^{-s}_X$ by the relation\n\\begin{equation}\\label{eq:Ixyz}\n\\mathcal{F}_Z I_{XY}(Z) \\mathcal{F}_Z^{-1} \\equiv \\int_Z^\\oplus I_{XY}^Z(k) \\text{d} k.\n\\end{equation}\nThe following fact is known: a continuous operator\n$T:\\mathcal{H}^t_Y\\to\\mathcal{H}^{-s}_X$ is of the preceding form if and only if\n$U_aT=TUa$ for all $a\\in Z$. From the preceding results we get\n(notations are as in Remark \\ref{re:iaff}):\n\n\\begin{proposition}\\label{pr:zxy}\n  Let $X,Y,Z\\in\\mathcal{S}$ with $Z\\subset X\\cap Y$ and assume that\n  $\\mathcal{G}^1_X=\\mathcal{H}^s_X$ and $\\mathcal{G}^1_Y=\\mathcal{H}^t_Y$. An operator\n  $I_{XY}(Z):\\mathcal{H}^t_Y\\to\\mathcal{H}^{-s}_X$ satisfies the conditions of\n  Remark \\ref{re:iaff} if and only if it is of the form\n  \\eqref{eq:Ixyz} with a norm continuous function $I_{XY}^Z:Z\\to\n  L^\\circ(\\mathcal{H}^t_{Y\/Z},\\mathcal{H}^{-s}_{X\/Z})$ satisfying \\eqref{eq:Iest}.\n\\end{proposition}\n\n\n\\subsection{Auxiliary results}\n\\label{ss:lemma}\n\nIn this subsection we collect some useful technical results.  Let\n$\\mathcal{E},\\mathcal{F},\\mathcal{G},\\mathcal{H}$ be Hilbert spaces. Note that we have a canonical\nidentification (tensor products are discussed in \\S\\ref{ss:ha})\n\\begin{equation}\\label{eq:comtens}\nK(\\mathcal{E},\\mathcal{F})\\otimes K(\\mathcal{G},\\mathcal{H})\\cong K(\\mathcal{E}\\otimes\\mathcal{G},\\mathcal{F}\\otimes\\mathcal{H}),\n\\hspace{2mm}\\text{in particular}\\hspace{2mm}\nK(\\mathcal{E},\\mathcal{F}\\otimes\\mathcal{H})\\cong K(\\mathcal{E},\\mathcal{F})\\otimes\\mathcal{H}.\n\\end{equation}\nAssume that we have continuous injective embeddings $\\mathcal{E}\\subset\\mathcal{G}$\nand $\\mathcal{F}\\subset\\mathcal{G}$. We equip $\\mathcal{E}\\cap\\mathcal{F}$ with the intersection\ntopology defined by the norm $(\\|g\\|_\\mathcal{E}^2+\\|g\\|_\\mathcal{F}^2)^{1\/2}$, hence\n$\\mathcal{E}\\cap\\mathcal{F}$ becomes a Hilbert space continuously embedded in $\\mathcal{G}$.\n\n\\begin{lemma}\\label{lm:efgh}\n  The map $K(\\mathcal{E},\\mathcal{H})\\times K(\\mathcal{F},\\mathcal{H}) \\to K(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})$ which\n  associates to $S\\in K(\\mathcal{E},\\mathcal{H})$ and $T\\in K(\\mathcal{F},\\mathcal{H})$ the operator\n  $S|_{\\mathcal{E}\\cap\\mathcal{F}}+T|_{\\mathcal{E}\\cap\\mathcal{F}} \\in K(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})$ is\n  surjective. Thus, slightly formally,\n\\begin{equation}\\label{eq:efgh}\nK(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})=K(\\mathcal{E},\\mathcal{H}) + K(\\mathcal{F},\\mathcal{H}).\n\\end{equation}\n\\end{lemma}\n\\proof It is clear that the map is well defined. Let $R\\in\nK(\\mathcal{E}\\cap\\mathcal{F},\\mathcal{H})$, we have to show that there are $S,T$ as in the\nstatement of the proposition such that $R=\nS|_{\\mathcal{E}\\cap\\mathcal{F}}+T|_{\\mathcal{E}\\cap\\mathcal{F}}$.  Observe that the norm on\n$\\mathcal{E}\\cap\\mathcal{F}$ has been chosen such that the linear map\n$g\\mapsto(g,g)\\in\\mathcal{E}\\oplus\\mathcal{F}$ be an isometry with range a closed\nlinear subspace $\\mathcal{I}$. Consider $R$ as a linear map $\\mathcal{I}\\to\\mathcal{H}$ and\nextend it to the orthogonal of $\\mathcal{I}$ by zero. The so defined map\n$\\widetilde R:\\mathcal{I}\\to\\mathcal{H}$ is clearly compact. Let $S,T$ be defined by\n$Se=\\widetilde R(e,0)$ and $Tf=\\widetilde R(0,f)$. Clearly $S\\in\nK(\\mathcal{E},\\mathcal{H})$ and $T\\in K(\\mathcal{F},\\mathcal{H})$ and if $g\\in\\mathcal{E}\\cap\\mathcal{F}$ then\n\\[\nSg+Tg=\\widetilde R(g,0)+\\widetilde R(0,g)=\\widetilde R(g,g)=Rg\n\\]\nwhich proves the lemma.\n\\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nWe give some applications. If $E,F$ are Euclidean spaces and $s>0$\nis real then\n\\begin{equation}\\label{eq:EFs}\n\\mathcal{H}^s_{E\\oplus F}=\\big(\\mathcal{H}^s_E\\otimes\\mathcal{H}_F\\big)\\cap\n\\big(\\mathcal{H}_E\\otimes\\mathcal{H}^s_F\\big)\n\\end{equation}\nhence Lemma \\ref{lm:efgh} gives for an arbitrary Hilbert space $\\mathcal{H}$\n\\begin{equation}\\label{eq:EFH}\nK(\\mathcal{H}^s_{E\\oplus F},\\mathcal{H})=\nK(\\mathcal{H}^s_E\\otimes\\mathcal{H}_F,\\mathcal{H}) + K(\\mathcal{H}_E\\otimes\\mathcal{H}^s_F,\\mathcal{H}).\n\\end{equation}\nIf $\\mathcal{H}$ itself is a tensor product $\\mathcal{H}=\\mathcal{H}'\\otimes\\mathcal{H}''$ then we\ncan combine this with  \\eqref{eq:comtens} and get\n\\begin{equation}\\label{eq:EFHEF}\nK(\\mathcal{H}^s_{E\\oplus F},\\mathcal{H}'\\otimes\\mathcal{H}'')  =\nK(\\mathcal{H}^s_E,\\mathcal{H}')\\otimes K(\\mathcal{H}_F,\\mathcal{H}'')\n + K(\\mathcal{H}_E,\\mathcal{H}') \\otimes K(\\mathcal{H}^s_F,\\mathcal{H}'').\n\\end{equation}\nConsider now a triplet $X,Y,Z$ such that $Z\\subset X\\cap Y$ and\ndenote\n\\begin{equation}\\label{eq:xyze}\nE=(X\\cap Y)\/Z \\hspace{2mm}\\text{and}\\hspace{2mm} \nX \\boxplus Y = X\/Y\\times Y\/X.\n\\end{equation}\nThen $Y\/Z=E\\oplus(Y\/X) \\text{ and } X\/Z=E\\oplus(X\/Y)$ hence by using\n\\eqref{eq:EFHEF} we get for example \n\\begin{align}\n\\mathcal{H}_{Y\/Z} &=\\mathcal{H}_E\\otimes\\mathcal{H}_{Y\/X} \\text{ and \\ }\n\\mathcal{H}_{X\/Z} =\\mathcal{H}_E\\otimes\\mathcal{H}_{X\/Y} \\label{eq:new} \\\\\n\\mathcal{H}^2_{Y\/Z} &=\\big(\\mathcal{H}^2_E\\otimes\\mathcal{H}_{Y\/X}\\big)\\cap \n\\big(\\mathcal{H}_E\\otimes\\mathcal{H}^2_{Y\/X}\\big) \\label{eq:klea} \\\\\n\\mathcal{H}^{-2}_{X\/Z} &= \\mathcal{H}^{-2}_E\\otimes\\mathcal{H}_{X\/Y}\n+\\mathcal{H}_E\\otimes\\mathcal{H}^{-2}_{X\/Y}. \\label{eq:klean}\n\\end{align}\nBy using once again \\eqref{eq:EFHEF} and the notations introduced in\n\\eqref{eq:ikef}, we get\n\\begin{equation}\\label{eq:kde3}\n\\mathscr{K}^2_{X\/Z,Y\/Z} = \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X} + \n\\mathscr{K}_E\\otimes\\mathscr{K}^2_{X\/Y,Y\/X}.\n\\end{equation}\nWe identify a Hilbert-Schmidt operator with its kernel, so $L^2(X\n\\boxplus Y)\\subset \\mathscr{K}_{X\/Y,Y\/X}$ is the subspace of Hilbert-Schmidt\noperators. The we have a strict inclusion\n\\begin{equation} \\label{eq:kde9} \nL^2(X \\boxplus Y;\\mathscr{K}^2_E) \\subset \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X}\n\\end{equation}\n\n\n\\subsection{First order regularity conditions}\n\\label{ss:scs} \n\nIn the next two subsections we consider interactions as in\nProposition \\ref{pr:nrm} and give explicit conditions on the\n$I_{XY}^Z$ such that $H$ be of class $C^1_\\mathrm{u}(D)$. We recall that\nthe assumptions of Proposition \\ref{pr:nrm} can be stated as\nfollows: for all $Z\\subset X\\cap Y$\n\\begin{align}\n& I^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{X\/Z} \n\\hspace{2mm} \\text{is compact and satisfies}\\hspace{2mm} \n(I^{Z}_{XY})^*\\supset I^Z_{YX}, \\label{eq:A}\\\\\n& [D,I^Z_{XY}]:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\hspace{2mm}\\text{is\ncompact}. \\label{eq:B}\n\\end{align}\nIf \\eqref{eq:A} is satisfied then by duality and interpolation we\nget\n\\begin{equation}\\label{eq:interpol}\nI^Z_{XY}:\\mathcal{H}^\\theta_{Y\/Z}\\to\\mathcal{H}^{\\theta-2}_{X\/Z} \\quad \n\\text{is a compact operator for all } 0\\leq \\theta \\leq 2,\n\\end{equation} \nin particular the operator $[D,I^Z_{XY}]\\equiv\nD_{X\/Z}I^Z_{XY}-I^Z_{XY}D_{Y\/Z}$ restricted to the space of\nfunctions in $\\mathcal{H}^2_{Y\/Z}$ with compact support has values in the\nspace of functions locally in $\\mathcal{H}^{-1}_{X\/Z}$.  We use, for\nexample, the relation $D_{X\/Z}=D_E\\otimes 1 + 1\\otimes D_{X\/Y}$\nrelatively to \\eqref{eq:new} to decompose this operator as follows:\n\\begin{align}\n[D,I^Z_{XY}] \n&=(D_E+D_{X\/Y})I^Z_{XY}-I^Z_{XY}(D_E+D_{Y\/X}) \\nonumber \\\\\n&=[D_E,I^Z_{XY}] +D_{X\/Y}I^Z_{XY} -I^Z_{XY}D_{Y\/X}. \\label{eq:dec}\n\\end{align}\nSince $I^Z_{XY}D_{Y\/X}\\subset (D_{Y\/X}I^Z_{YX})^*$ if \\eqref{eq:A}\nis satisfied then condition \\eqref{eq:B} follows from:\n\\begin{equation}\\label{eq:dii}\n[D_E,I^Z_{XY}] \\text{ and } D_{X\/Y}I^Z_{XY} \\text{ are compact\n  operators } \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z} \\text{ for all } X,Y,Z.\n\\end{equation}\nAccording to \\eqref{eq:kde3} the first part of condition \n\\eqref{eq:A} is equivalent to\n\\begin{equation}\\label{eq:kde4}\nI_{XY}^Z=J+J' \\text{ for some }\nJ\\in \\mathscr{K}^2_E\\otimes \\mathscr{K}_{X\/Y,Y\/X} \\text{ and }\nJ'\\in\\mathscr{K}_E\\otimes\\mathscr{K}^2_{X\/Y,Y\/X}. \n\\end{equation}\nAs a particular case, from \\eqref{eq:kde9} we obtain the example\ndiscussed in \\S\\ref{ss:examples}. The compactness conditions\n\\eqref{eq:dii} are conditions on the kernels $[D_E,I^Z_{XY}(x',y')]$\nand $x'\\cdot\\nabla_{x'}I_{XY}^Z(x',y')$ of the operators\n$[D_E,I^Z_{XY}]$ and $D_{X\/Y}I^Z_{XY}$.  Note that a condition on\n$I^Z_{XY}D_{Y\/X}$ is a requirement on the kernel\n$y'\\cdot\\nabla_{y'}I_{XY}^Z(x',y')$.\n\n\n\\subsection{Creation-annihilation type interactions}\n\\label{ss:xsupy} \n\nTo see the relation with the\ncreation-annihilation type interactions characteristic to quantum\nfield models we consider now the case when\n$Y\\subset X$ strictly. Then\n\\begin{equation}\\label{eq:ysux}\n\\mathscr{C}_{XY}=\\mathscr{C}_Y\\otimes\\mathcal{H}_{X\/Y}, \\quad \n\\mathscr{C}_{XY}(Z)=\\mathscr{C}_Y(Z)\\otimes\\mathcal{H}_{X\/Y}, \\quad\n\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}\n\\end{equation}\nwhere the first two tensor product have to be interpreted as\nexplained in \\S\\ref{ss:ha}. In particular we have\n\\begin{equation}\\label{eq:sux}\nL^2(X\/Y;\\mathscr{C}_Y)\\subset\\mathscr{C}_{XY} \\quad \\text{and} \\quad\nL^2(X\/Y;\\mathscr{C}_Y(Z))\\subset\\mathscr{C}_{XY}(Z)\n\\quad \\text{strictly}. \n\\end{equation}\nIf $Z\\subset Y$ then $X=Z\\oplus(Y\/Z)\\oplus(X\/Y)$\nand $X\/Z=(Y\/Z)\\oplus(X\/Y)$ hence $\\mathcal{H}_{X\/Z}=\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}$\nand thus the operator $I^Z_{XY}$ is\njust a compact operator\n\\begin{equation}\\label{eq:ixyo}\nI^Z_{XY} : \\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}.\n\\end{equation}\nIf $\\mathcal{E},\\mathcal{F},\\mathcal{G}$ are Hilbert spaces then $K(\\mathcal{E},\\mathcal{F}\\otimes\\mathcal{G})\\cong\nK(\\mathcal{E},\\mathcal{F})\\otimes\\mathcal{G}$. Hence \\eqref{eq:ixyo} means\n\\begin{equation}\\label{eq:ixyoo}\nI^Z_{XY} \\in \\mathscr{K}^2_{Y\/Z}\\otimes \\mathcal{H}_{X\/Y}.\n\\end{equation}\nLet $\\mathscr{I}_{XY}= {\\textstyle\\sum_{Z\\subset X\\cap Y}} 1_Z\\otimes\n\\mathscr{K}^2_{X\/Z,Y\/Z}$, where the sum is direct and closed in\n$\\mathscr{K}^2_{XY}$.  A usual nonrelativistic $N$-body Hamiltonian\nassociated to the semilattice $\\mathcal{S}_X$ of subspaces of $X$ is of the\nform $\\Delta_X+I_X$ with $I_X\\in\\mathscr{I}_{X}\\equiv\\mathscr{I}_{XX}$. Thus the\ninteraction which couples the $X$ and $Y$ systems is of the form\n\\begin{equation}\\label{eq:xyinter}\nI_{XY}={\\textstyle\\sum_{Z\\in\\mathcal{S}_Y}} \n1_Z\\otimes I^Z_{XY} \\in \\mathscr{I}_{Y}\\otimes \\mathcal{H}_{X\/Y}. \n\\end{equation}\nIn particular we may take $I_{XY}\\in L^2(X\/Y;\\mathscr{I}_Y)$, but we stress\nthat the space $\\mathscr{I}_{Y}\\otimes \\mathcal{H}_{X\/Y}$ is much larger (see\n\\S\\ref{ss:ha}).  More explicitly, a square integrable function\n$I_{XY}:X\/Y\\to\\mathscr{I}_Y$ determines an operator $I_{XY}:\\mathcal{H}^2_Y\\to\\mathcal{H}_X$\nby the following rule: it associates to $u\\in\\mathcal{H}^2(Y)$ the function\n$y'\\mapsto I_{XY}(y')u$ which belongs to $L^2(X\/Y;\\mathcal{H}_{X\/Y})=\\mathcal{H}_X$.\nWe may also write\n\\begin{equation}\\label{eq:IV}\n(I_{XY}u)(x)=(I_{XY}(y')u)(y) \\quad \\text{where }\nx\\in X=Y\\oplus X\/Y \\text{ is written as } x=(y,y').\n\\end{equation}\nWe say that the operator valued function $I_{XY}$ is the kernel of\nthe operator $I_{XY}$. The adjoint $I_{YX}=I_{XY}^*$ is an integral\noperator in the $y'$ variable (like an annihilation\noperator). Indeed, if $v\\in\\mathcal{H}_X$ is thought as a map $y'\\mapsto\nv(y')\\in\\mathcal{H}_Y$ then we have $I_{YX}v=\\int_{X\/Y}\nI^*_{XY}(y')v(y')\\text{d} y'$ at least formally.\n\n\nThe particular case when the function $I_{XY}$ is factorizable\nclarifies the connection with the quantum field type interactions:\nlet $I_{XY}$ be a finite sum $I_{XY}=\\sum_i V_Y^i\\otimes\\phi_i$\nwhere $V^i_Y\\in\\mathscr{I}_Y$ and $\\phi_i\\in \\mathcal{H}_{X\/Y}$, then\n\\begin{equation}\\label{eq:qft}\nI_{XY}u={\\textstyle\\sum_i} (V_Y^i u)\\otimes\\phi_i \\quad\n\\text{as an operator }  \nI_{XY}:\\mathcal{H}^2_Y\\to\\mathcal{H}_X=\\mathcal{H}_Y\\otimes\\mathcal{H}_{X\/Y}.\n\\end{equation}\nThis is a sum of $N$-body type interactions $V^i_Y$ tensorized with\noperators which create particles in states~ $\\phi_i$.\n\nThe conditions on the ``commutator'' $[D,I_{XY}]$ may be written\nin terms of the kernel of $I_{XY}$. The\nrelation \\eqref{eq:dec}  becomes $\n[D,I_{XY}]=[D_Y,I_{XY}]+D_{X\/Y}I_{XY}$. The operator $D_Y$ acts only\non the variable $y$ and $D_{X\/Y}$ acts only on the variable\n$y'$. Thus $[D_Y,I_{XY}]$ and $D_{X\/Y}I_{XY}$ are operators of the\nsame nature as $I_{XY}$ but more singular: the kernel of\n$[D_Y,I_{XY}]$ is the function $y'\\mapsto [D_Y,I_{XY}(y')]$ and that\nof $2iD_{X\/Y}I_{XY}$ is the function $ y'\\mapsto\n(y'\\cdot\\nabla_{y'}+n\/2) I_{XY}(y')$.  Thus, to get\n\\eqref{eq:B} it suffices to\nrequire two conditions on the kernel $I_{XY}$, one on\n$[D_Y,I_{XY}(y')]$ and a second one on\n$y'\\cdot\\nabla_{y'}I_{XY}(y')$.\n\nIf we decompose $I_{XY}$ as in \\eqref{eq:xyinter} with\n$I^Z_{XY}:\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}$ compact then the\n(formal) kernel of $I^Z_{XY}$ is a $\\mathscr{K}^2_{Y\/Z}$ valued map on\n$X\/Y$.  We require that $[D_{Y\/Z},I^Z_{XY}]$ and $D_{X\/Y}I^Z_{XY}$\nbe compact operators $\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}$. From\n\\eqref{eq:stens} and $X\/Z=(Y\/Z)\\oplus(X\/Y)$ we get\n\\begin{equation*}\n\\mathcal{H}^2_{X\/Z} = \\big(\\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}^2_{X\/Y}\\big)\\cap\n\\big(\\mathcal{H}^2_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}\\big), \\quad\n\\mathcal{H}^{-2}_{X\/Z} = \\mathcal{H}_{Y\/Z}\\otimes\\mathcal{H}^{-2}_{X\/Y} +\n\\mathcal{H}^{-2}_{Y\/Z}\\otimes\\mathcal{H}_{X\/Y}\n\\end{equation*}  \nwhich are helpful in checking these compactness requirements.\n\n\n\n\\subsection{Besov regularity classes}\n\\label{ss:scsc} \n\nWe recall some facts concerning the Besov type regularity class $\nC^{1,1}(D)$; we refer to \\cite{ABG} for details on these\nmatters. Since the conjugate operator $D$ is fixed we shall not\nindicate it in the notation from now on. An operator $T\\in L(\\mathcal{H})$\nis of class $C^{1,1}$ if\n\\begin{equation}\\label{eq:c11}\n\\int_0^1\\|W^*_{2\\varepsilon}T W_{2\\varepsilon}-\n2W^*_{\\varepsilon}T W_{\\varepsilon} +T\\|\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} \\equiv\n\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2\nT\\|\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} <\\infty\n\\end{equation}\nwhere $\\mathcal{W}_\\varepsilon$ is the automorphism of $L(\\mathcal{H})$ defined by\n$\\mathcal{W}_\\varepsilon T=W_\\varepsilon^*T W_\\varepsilon$. The condition\n\\eqref{eq:c11} implies that $T$ is of class $C^1_\\mathrm{u}$ and is\njust slightly more than this. Indeed, $T$ is of class $C^1$ or\n$C^1_\\mathrm{u}$ if and only if the limit\n\\[\n\\lim_{\\tau\\to 0}\\int_\\tau^1 (\\mathcal{W}_\\varepsilon-1)^2 T\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2}\n\\] \nexists strongly or in norm respectively. The following subclass of\n$C^{1,1}$ is useful in applications:  $T$ is called of class\n$C^{1+}$ if $T$ is of class $C^1$, so the commutator $[D,T]$ is a\nbounded operator, and\n\\begin{equation}\\label{eq:din}\n\\int_0^1\\|W_\\varepsilon^*[D,T]W_\\varepsilon-[D,T]\\|\n\\frac{\\text{d}\\varepsilon}{\\varepsilon} <\\infty.\n\\end{equation}\nThen $C^{1+}\\subset C^{1,1}$. The class most frequently used in the\ncontext of the Mourre theorem is $C^2$: this is the set of $T\\in\nC^1$ such that $[D,T]\\in C^1$. But $[D,T]\\in C^1$ if and only if\n\\[\n\\|W_\\varepsilon^*[D,T] W_\\varepsilon-[D,T]\\| \\leq \nC |\\varepsilon| \\quad \\text{for some constant $C$ and all real }\n\\varepsilon \n\\]\nhence this condition is much stronger then the Dini type condition\n\\eqref{eq:din}.  A self-adjoint operator $H$ is of class $C^{1,1}$,\n$C^{1+}$ or $C^2$ if its resolvent is of class $C^{1,1}$, $C^{1+}$\nor $C^2$ respectively. \n\nWe now consider a Hamiltonian as in Proposition \\ref{pr:nrm} and\ndiscuss conditions which ensure that $H$ is of class $C^{1,1}$. An\nimportant point is that the domain $\\mathcal{H}^2$ of $H$ is stable under\nthe dilation group $W_\\tau$.  Then Theorem 6.3.4 from \\cite{ABG}\nimplies that $H$ is of class $C^{1,1}$ if and only if\n\\begin{equation}\\label{eq:c11h}\n\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2H\\|_{\\mathcal{H}^2\\to\\mathcal{H}^{-2}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} <\\infty.\n\\end{equation}\nAs above $\\mathcal{W}_{\\varepsilon} H=W^*_{\\varepsilon}H W_{\\varepsilon}$\nhence $ (\\mathcal{W}_\\varepsilon-1)^2H=W^*_{2\\varepsilon}H W_{2\\varepsilon}-\n2W^*_{\\varepsilon}H W_{\\varepsilon} +H$.  We have $H=\\Delta +I$ and\ndue to \\eqref{eq:dlap} the relation \\eqref{eq:c11h} is trivially\nverified by the kinetic part $\\Delta$ of $H$ hence we are only\ninterested in conditions on $I$ which ensure that \\eqref{eq:c11h} is\nsatisfied with $H$ replaced by $I$. If this is the case, by a slight\nabuse of language we say that $I$ is of class $C^{1,1}$. In terms of\nthe coefficients $I_{XY}$, this means\n\\begin{equation}\\label{eq:c11xyz}\n\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2I_{XY}^Z\\|_{\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} <\\infty\n\\quad\\text{for all } X,Y,Z.\n\\end{equation}\nWe recall one fact (see \\cite[Ch. 5]{ABG}). Let $I:\\mathcal{H}^2\\to\\mathcal{H}^{-2}$\nbe an arbitrary linear continuous operator. Then\n$[D,I]:\\mathcal{H}^2_\\mathrm{c}\\to\\mathcal{H}^{-3}_{\\mathrm{loc}}$ is well defined and $I$\nis of class $C^1$ (in an obvious sense) if and only if this operator\nis the restriction of a continuous map $\\mathcal{H}^2\\to\\mathcal{H}^{-2}$, which\nwill be denoted also $[D,I]$. We say that $I$ is of class $C^{1+}$\nif this condition is satisfied and \n\\begin{equation}\\label{eq:dini}\n\\int_0^1\\|W_\\varepsilon^*[D,I]W_\\varepsilon-[D,I]\\|_{\\mathcal{H}^2\\to\\mathcal{H}^{-2}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon} <\\infty.\n\\end{equation}\nAs before, if $I$ is of class $C^{1+}$ then it is of class\n$C^{1,1}$. In terms of the coefficients $I_{XY}^Z$ this means\n\\begin{equation}\\label{eq:dinix}\n\\int_0^1\\|W_\\varepsilon^*[D,I_{XY}^Z]W_\\varepsilon-[D,I_{XY}^Z]\n\\|_{\\mathcal{H}^2_{Y\/Z}\\to\\mathcal{H}^{-2}_{X\/Z}}\n\\frac{\\text{d}\\varepsilon}{\\varepsilon} <\\infty.\n\\end{equation}\nSuch a condition should be imposed on each of the three terms in the\ndecomposition \\eqref{eq:dec} separately.\n\nThe techniques developed in \\S 7.5.3 and on pages 425--429 from\n\\cite{ABG} can be used to get more concrete conditions.  The only\nnew fact with respect to the $N$-body situation as treated there is\nthat $\\mathcal{W}_\\tau$ when considered as an operator on $\\mathscr{L}_{X\/Z,Y,Z}$\nfactorizes in a product of three commuting operators. Indeed, if we\nwrite $\\mathcal{H}_{Y\/Z}=\\mathcal{H}_E\\otimes\\mathcal{H}_{Y\/X}$ and\n$\\mathcal{H}_{X\/Z}=\\mathcal{H}_E\\otimes\\mathcal{H}_{X\/Y}$ then we get\n$\\mathcal{W}_\\tau(T)=W^{X\/Y}_{-\\tau}\\mathcal{W}^E_\\tau(T)W^{Y\/X}_\\tau$ where this\ntime we indicated by an upper index the space to which the operator\nis related and, for example, we identified $W^{Y\/X}_\\tau=1_E\\otimes\nW^{Y\/X}_\\tau$. To check the $C^{1,1}$ property in this context\none may use:\n\n\n\\begin{proposition}\\label{pr:inter}\nIf $T\\in\\mathscr{L}:= L(\\mathcal{H}^2_{Y\/Z},\\mathcal{H}^{-2}_{X\/Z})$ then \n$\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2T\\|_\\mathscr{L}\n\\text{d}\\varepsilon\/\\varepsilon^2<\\infty$ follows from\n\\begin{equation}\\label{eq:inter}\n\\int_0^1\\left(\n\\|(W^{X\/Y}_{\\varepsilon}-1)^2T\\|_\\mathscr{L}+ \n\\|(\\mathcal{W}^E_\\varepsilon-1)^2T\\|_\\mathscr{L}+\n\\|T(W^{Y\/X}_{\\varepsilon}-1)^2\\|_\\mathscr{L}\n\\right)\n\\frac{\\text{d}\\varepsilon}{\\varepsilon^2} < \\infty.\n\\end{equation}\n\\end{proposition}\n\\proof We shall interpret $\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2T\\|_\\mathscr{L}\n\\text{d}\\varepsilon\/\\varepsilon^2<\\infty$ in terms of real interpolation\ntheory. Let $L_\\tau$ be the operator of left multiplication by\n$W^{X\/Y}_{-\\tau}$ and $N_\\tau$ the operator of right multiplication\nby $W^{Y\/X}_{\\tau}$ on $\\mathscr{L}_{X\/Z,Y\/Z}$. If we also set\n$M_\\tau=\\mathcal{W}^E_\\tau$ then we get three commuting operators\n$L_\\tau,M_\\tau,N\\tau$ on $\\mathscr{L}_{X\/Z,Y\/Z}$ such that $\\mathcal{W}_\\tau=L_\\tau\nM_\\tau N_\\tau$.  Then it is easy to check a Dini type condition like\n\\eqref{eq:dinix} by using\n\\begin{equation}\\label{eq:tmp}\n\\mathcal{W}_\\tau-1=(L_\\tau-1)M_\\tau N_\\tau+(M_\\tau -1)N_\\tau+(N_\\tau-1). \n\\end{equation}\nOn the other hand, observe that $\\mathcal{W}_\\tau,L_\\tau,M_\\tau,N_\\tau$ are\none parameter groups of operators on the Banach space $\\mathscr{L}$.  These\ngroups are not continuous in the ordinary sense but this does not\nreally matter, in fact we are in the setting of \\cite[Ch. 5]{ABG}.\nThe main point is that the integral\n$\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^2T\\|_\\mathscr{L}\\text{d}\\varepsilon\/\\varepsilon^2$\nis finite if and only if\n$\\int_0^1\\|(\\mathcal{W}_\\varepsilon-1)^6T\\|_\\mathscr{L}\\text{d}\\varepsilon\/\\varepsilon^2$\nis finite (see Theorem 3.4.6 in \\cite{ABG}; this is where real\ninterpolation comes into play).  Now by taking the sixth power of\n\\eqref{eq:tmp} and developing the right hand side we easily get the\nresult, cf. the formula on top of page 132 of \\cite{ABG}.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nThe proof of Theorem \\ref{th:BVR} is based on an extension of\nPropositions 9.4.11 and 9.4.12 from \\cite{ABG} to the present\ncontext. Since the argument is very similar, we do not enter into\ndetails. We mention only that the operator $D$ can be written as\n$4D=P \\cdot Q+Q \\cdot P$ where $P=\\oplus_X P_X$ and $Q=\\oplus_X Q_X$\nare suitably interpreted. The proofs in \\cite{ABG} depend only on\nthis structure.\n\n\n \n\n\\section{Appendix: Hamiltonian algebras} \n\\label{s:appb}\n\n\nWe prove here some results on $C^*$-algebras generated by certain\nclasses of ``elementary'' Hamiltonians.\n\n\\subsection{}\\label{ss:a1}\n\nLet $X$ be a locally compact abelian group and let\n$\\{U_x\\}_{x\\in X}$ be a strongly continuous unitary representation\nof $X$ on a Hilbert space $\\mathcal{H}$. Then one can associate to it a\nBorel regular spectral measure $E$ on $X^*$ with values projectors\non $\\mathcal{H}$ such that $U_x=\\int_{X^*}k(x)E(\\text{d} k)$ and this allows us\nto define for each Borel function $\\psi:X^*\\to\\mathbb{C}$ a normal\noperator on $\\mathcal{H}$ by the formula $\\psi(P)=\\int_{X^*} \\psi(k)E(\\text{d}\nk)$.  The set $\\mathscr{T}_X(\\mathcal{H})$ of all the operators $\\psi(P)$ with\n$\\psi\\in\\cc_{\\mathrm{o}}(X^*)$ is clearly a non-degenerate $C^*$-algebra of\noperators on $\\mathcal{H}$. The following result, which will be useful in\nseveral contexts, is an easy consequence of the Cohen-Hewitt\nfactorization theorem, see Lemma 3.8 from \\cite{GI4}. Consider an\noperator $A\\in L(\\mathcal{H})$.\n\n\\begin{lemma}\\label{lm:help}\n$\\displaystyle{\\lim_{x\\to0}}\\|(U_x-1)A\\|=0$ if and\nonly if $A=\\psi(P)B$ for some $\\psi\\in\\cc_{\\mathrm{o}}(X^*)$ and $B\\in L(\\mathcal{H})$.\n\\end{lemma}\n\nWe say that an operator $S\\in L(\\mathcal{H})$ is of class $C^0(P)$ if the\nmap $x\\mapsto U_xSU_x^*$ is norm continuous.\n\n\\begin{lemma}\\label{lm:cop}\nLet $S\\in L(\\mathcal{H})$ be of class $C^0(P)$ and let $T\\in\n\\mathscr{T}_X(\\mathcal{H})$. Then for each $\\varepsilon>0$ there is $Y\\subset X$\nfinite and there are operators $T_y\\in \\mathscr{T}_X(\\mathcal{H})$ such that\n$\\|ST-\\sum_{y\\in Y} T_y U_{y}SU_{y}^*\\|<\\varepsilon$.\n\\end{lemma}\n\\proof It suffices to assume that $T=\\psi(P)$ where $\\psi$ has a\nFourier transform integrable on $X$, so that $T=\\int_X U_x\n\\widehat\\psi(x) \\text{d} x$, and then to use a partition of unity on $X$\nand the uniform continuity of the map $x\\mapsto U_xSU_x^*$ (see the\nproof of Lemma 2.1 in \\cite{DG1}).  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\nWe say that a subset $\\mathcal{B}$ of $L(\\mathcal{H})$ is $X$-stable if\n$U_xSU_x^*\\in\\mathcal{B}$ whenever $S\\in\\mathcal{B}$ and $x\\in X$. From Lemma\n\\ref{lm:cop} we see that if $\\mathcal{B}$ is an $X$-stable real linear space\nof operators of class $C^0(P)$ then\n\\[\n\\mathcal{B}\\cdot \\mathscr{T}_X(\\mathcal{H})= \\mathscr{T}_X(\\mathcal{H})\\cdot\\mathcal{B}.\n\\] \nSince the $C^*$-algebra $\\mathcal{A}$ generated by $\\mathcal{B}$ is also $X$-stable\nand consists of operators of class $C^0(P)$\n\\begin{equation}\\label{eq:cop}\n\\mathscr{A}\\equiv\\mathcal{A}\\cdot \\mathscr{T}_X(\\mathcal{H})= \\mathscr{T}_X(\\mathcal{H})\\cdot\\mathcal{A}\n\\end{equation} \nis a $C^*$-algebra. The operators $U_x$ implement a norm continuous\naction of $X$ by automorphisms of the algebra $\\mathcal{A}$ so the\n$C^*$-algebra crossed product $\\mathcal{A}\\rtimes X$ is well defined and the\nalgebra $\\mathscr{A}$ is a quotient of this crossed product.\n\nA function $h$ on $X^*$ is called \\emph{$p$-periodic} for some\nnon-zero $p\\in X^*$ if $h(k+p)=h(k)$ for all $k\\in X^*$.\n\n\\begin{proposition}\\label{pr:cop}\nLet $\\mathcal{V}$ be an $X$-stable set of symmetric bounded operators of\nclass $C^0(P)$ and such that $\\lambda\\mathcal{V}\\subset\\mathcal{V}$ if\n$\\lambda\\in\\mathbb{R}$. Denote $\\mathcal{A}$ the $C^*$-algebra generated by $\\mathcal{V}$\nand define $\\mathscr{A}$ by \\eqref{eq:cop}.  Let $h:X^*\\to\\mathbb{R}$ be\ncontinuous, not $p$-periodic if $p\\neq0$, and such that\n$|h(k)|\\to\\infty$ as $k\\to\\infty$.  Then $\\mathscr{A}$ is the $C^*$-algebra\ngenerated by the self-adjoint operators of the form $h(P+k)+V$ with\n$k\\in X^*$ and $V\\in\\mathcal{V}$.\n\\end{proposition}\n\\proof Denote $K=h(P+k)$ and let $R_\\lambda=(z-K-\\lambda V)^{-1}$\nwith $z$ not real and $\\lambda$ real. Let $\\mathscr{C}$ be the $C^*$-algebra\ngenerated by such operators (with varying $k$ and $V$). By taking\n$V=0$ we see that $\\mathscr{C}$ will contain the $C^*$-algebra generated by\nthe operators $R_0$. By the Stone-Weierstrass theorem this algebra\nis $\\mathscr{T}_X(\\mathcal{H})$ because the set of functions $p\\to(z-h(p+k))^{-1}$\nwhere $k$ runs over $X^*$ separates the points of $X^*$. The\nderivative with respect to $\\lambda$ at $\\lambda=0$ of $R_\\lambda$\nexists in norm and is equal to $R_0VR_0$, so $R_0VR_0\\in\\mathscr{C}$. Since\n$\\mathscr{T}_X\\subset\\mathscr{C}$ we get $\\phi(P)V\\psi(P)\\in\\mathscr{C}$ for all\n$\\phi,\\psi\\in\\cc_{\\mathrm{o}}(X^*)$ and all $V\\in\\mathcal{V}$. Since $V$ is of class\n$C^0(P)$ we have $(U_x-1)V\\psi(P)\\sim V(U_x-1)\\psi(P)\\to0$ in norm\nas $x\\to0$ from which we get $\\phi(P)V\\psi(P)\\to S\\psi(P)$ in norm\nas $\\phi\\to1$ conveniently. Thus $V\\psi(P)\\in\\mathscr{C}$ for $V,\\psi$ as\nabove. This implies $V_1\\cdots V_n\\psi(P)\\in\\mathscr{C}$ for all\n$V_1,\\dots,V_n\\in\\mathcal{V}$. Indeed, assuming $n=2$ for simplicity, we\nwrite $\\psi=\\psi_1\\psi_2$ with $\\psi_i\\in\\cc_{\\mathrm{o}}(X^*)$ and then Lemma\n\\ref{lm:cop} allows us to approximate $V_2\\psi_1(P)$ in norm with\nlinear combinations of operators of the form $\\phi(P)V^x_2$ where\nthe $V^x_2$ are translates of $V_2$.  Since $\\mathscr{C}$ is an algebra we\nget $V_1\\phi(P) V^x_2\\psi_2(P)\\in\\mathscr{C}$ hence passing to the limit we\nget $V_1V_2\\psi(P)\\in\\mathscr{C}$. Thus we proved $\\mathscr{A}\\subset\\mathscr{C}$. The\nconverse inclusion follows from a series expansion of $R_\\lambda$ in\npowers of $V$.  \\hfill \\vrule width 8pt height 9pt depth-1pt \\medskip\n\n\nThe next two corollaries follow easily from Proposition\n\\ref{pr:cop}. We take $\\mathcal{H}=L^2(X)$ which is equipped with the usual\nrepresentations $U_x,V_k$ of $X$ and $X^*$ respectively. Let\n$W_\\xi=U_xV_k$ with $\\xi=(x,k)$ be the phase space translation\noperator, so that $\\{W_\\xi\\}$ is a projective representation of the\nphase space $\\Xi=X\\oplus X^*$.  Fix some classical kinetic energy\nfunction $h$ as in the statement of Proposition \\ref{pr:cop} and let\nthe classical potential $v:X\\to\\mathbb{R}$ be a bounded uniformly\ncontinuous function. Then the quantum Hamiltonian will be\n$H=h(P)+v(Q)\\equiv K+V$. Since the origins in the configuration and\nmomentum spaces $X$ and $X^*$ have no special physical meaning one\nmay argue \\cite{Be1,Be2} that $W_\\xi H W^*_\\xi=h(P-k)+v(Q+x)$ is a\nHamiltonian as good as $H$ for the description of the evolution of\nthe system. It is not clear to us whether the algebra generated by\nsuch Hamiltonians (with $h$ and $v$ fixed) is in a natural way a\ncrossed product. On the other hand, it is natural to say that the\ncoupling constant in front of the potential is also a variable of\nthe system and so the Hamiltonians $H_\\lambda=K+\\lambda V$ with any\nreal $\\lambda$ are as relevant as $H$. Then we may apply Proposition\n\\ref{pr:cop} with $\\mathcal{V}$ equal to the set of operators of the form\n$\\lambda v(Q+x)$. Thus:\n\n\n\\begin{corollary}\\label{co:cop1}\nLet $v\\in\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ real and let $\\mathcal{A}$ be the $C^*$-subalgebra of\n$\\cc_{\\mathrm{b}}^{\\mathrm{u}}(X)$ generated by the translates of $v$.  Let $h:X^*\\to\\mathbb{R}$ be\ncontinuous, not $p$-periodic if $p\\neq0$, and such that\n$|h(k)|\\to\\infty$ as $k\\to\\infty$.  Then the $C^*$-algebra generated\nby the self-adjoint operators of the form $W_\\xi H_\\lambda W^*_\\xi$\nwith $\\xi\\in\\Xi$ and real $\\lambda$ is the crossed product\n$\\mathcal{A}\\rtimes X$.\n\\end{corollary}\n\nNow let $\\mathcal{T}$ be a set of closed subgroups of $X$ such that the\nsemilattice $\\mathcal{S}$ generated by it (i.e. the set of finite\nintersections of elements of $\\mathcal{T}$) consists of pairwise compatible\nsubgroups. Set $\\mathcal{C}_X(\\mathcal{S})=\\sum^\\mathrm{c}_{Y\\in\\mathcal{S}} \\mathcal{C}_X(Y)$. From\n\\eqref{eq:reg1} it follows that this is the $C^*$-algebra generated\nby  $\\sum_{Y\\in\\mathcal{T}} \\mathcal{C}_X(Y)$.\n\n\\begin{corollary}\\label{co:cop2}\nLet $h$ be as in Corollary \\ref{co:cop1}. Then the $C^*$-algebra\ngenerated by the self-adjoint operators of the form $h(P+k)+v(Q)$\nwith $k\\in X^*$ and $v\\in\\sum_{Y\\in\\mathcal{T}} \\mathcal{C}_X(Y)$ is the\ncrossed product $\\mathcal{C}_X(\\mathcal{S})\\rtimes X$.\n\\end{corollary}\n\n\n\\begin{remark}\\label{re:cop}\nProposition \\ref{pr:cop} and Corollaries \\ref{co:cop1} and\n\\ref{co:cop2} remain true and are easier to prove if we consider the\n$C^*$-algebra generated by the operators $h(P)+V$ with all\n$h:X^*\\to\\mathbb{R}$ continuous and such that $|h(k)|\\to\\infty$ as\n$k\\to\\infty$. If in Proposition \\ref{pr:cop} we take $\\mathcal{H}=L^2(X;E)$\nwith $E$ a finite dimensional Hilbert space (describing the spin\ndegrees of freedom) then the operators $H_0=h(P)$ with $h:X\\to L(E)$\na continuous symmetric operator valued function such that\n$\\|(h(k)+i)^{-1}\\|\\to 0$ as $k\\to\\infty$ are affiliated to $\\mathscr{A}$\nhence also their perturbations $H_0+V$ where $V$ satisfies the\ncriteria from \\cite{DG3}, for example.\n\\end{remark}\n\n\n\\subsection{}\n\\label{ss:anbody} \n\nWe consider the framework of \\S\\ref{ss:cexample} and use Corollary\n\\ref{co:cop2} to prove that the Hamiltonian algebra of a\nnonrelativistic $N$-body system is generated in a natural way by the\noperators of the form \\eqref{eq:nonrel}. To state a precise result\nit suffices to consider the reduced Hamiltonians (for which we keep\nthe notation $H$).\n\nLet $\\mathfrak{S}_2$ be the set of cluster decompositions which\ncontain only one nontrivial cluster which consists of exactly two\nelements. This cluster is of the form $\\{j,k\\}$ for a unique pair of\nnumbers $1\\leq j < k \\leq N$ and we denote by $(jk)$ the\ncorresponding cluster decomposition. The map $x\\mapsto x_j-x_k$\nsends $X$ onto $\\mathbb{R}^d$ and has $X_{(jk)}$ as kernel hence\n$V_{jk}(x_j-x_k)=V_{(jk)}\\circ\\pi_{(jk)}(x)$ where\n$V_{(jk)}:X\/X_{(jk)}\\to\\mathbb{R}$ is continuous with compact support and\n$\\pi_{(jk)}:X\\to X\/X_{(jk)}$ is the canonical surjection.\n\nThus the reduced Hamiltonians corresponding to \\eqref{eq:nonrel} are\nthe operators on $\\mathcal{H}_X$ of the form\n\\begin{equation}\\label{eq:nonre}\n\\Delta_X+\n{\\textstyle{\\sum_{\\sigma\\in\\mathfrak{S}_2}}} V_\\sigma\\circ\\pi_\\sigma\n\\hspace{2mm}\\text{with}\\hspace{2mm} V_\\sigma:X\/X_\\sigma\\to\\mathbb{R}\n\\text{ continuous with compact support}.\n\\end{equation} \nThese operators must be affiliated to the Hamiltonian algebra of the\n$N$-body system. On the other hand, if a Hamiltonian $h(P)+V$ is\nconsidered as physically admissible then $h(P+k)+V$ should be\nadmissible too because the zero momentum $k=0$ should not play a\nspecial role. In other terms, translations in momentum space should\nleave invariant the set of admissible Hamiltonians. Hence it is\nnatural to consider \\emph{the smallest $C^*$-algebra $\\mathscr{C}_X(\\mathcal{S})$\n  such that the operators \\eqref{eq:nonre} are affiliated to it and\n  which is stable under translations in momentum space. But this\n  algebra is exactly the crossed product}\n\\[\n\\mathscr{C}_X=\\mathcal{C}_X\\rtimes X=\\mathcal{C}_X\\cdot \\mathscr{T}_X \n\\hspace{2mm}\\text{with}\\hspace{2mm}\n\\mathcal{C}_X={\\textstyle\\sum_\\sigma}\\mathscr{C}_X(X_\\sigma).\n\\]\nIndeed, it is clear that the semilattice generated by\n$\\mathfrak{S}_2$ is $\\mathfrak{S}$ so we may apply Corollary\n\\ref{co:cop2}.\n\n\n\\subsection{}\n\\label{ss:amotiv} \n\nHere we prove Theorem \\ref{th:motiv}.\n\nLet $\\mathscr{C}'$ be the $C^*$-algebra generated by the operators of the\nform $(z-K-\\phi)^{-1}$ where $z$ is a not real number, $K$ is a\nstandard kinetic energy operator, and $\\phi$ is a symmetric field\noperator. With the notation \\eqref{eq:d} we easily get\n$\\mathscr{T}_\\text{d}\\subset\\mathscr{C}'$.  If $\\lambda\\in\\mathbb{R}$ then $\\lambda\\phi$ is\nalso a field operator so $(z-K-\\lambda\\phi)^{-1}\\in\\mathscr{C}'$. By taking\nthe derivative with respect to $\\lambda$ at $\\lambda=0$ of this\noperator we get $(z-K)^{-1}\\phi (z-K)^{-1}\\in\\mathscr{C}$. Since\n$(z-K)^{-1}=\\oplus_X(z-h_X(P))^{-1}$ (recall that $P$ is the\nmomentum observable independently of the group $X$) and since\n$\\mathscr{T}_\\text{d}\\subset\\mathscr{C}'$ we get $S\\phi(\\theta) T\\in\\mathscr{C}'$ for all\n$S,T\\in \\mathscr{T}_\\text{d}$ and $\\theta=(\\theta_{XY})_{X\\supset Y}$.\n\nLet $\\mathscr{C}'_{XY}=\\Pi_X\\mathscr{C}'\\Pi_Y\\subset\\mathscr{L}_{XY}$ be the components of\nthe algebra $\\mathscr{C}'$ and let us fix $X\\supset Y$. Then we get\n$\\varphi(P) a^*(u) \\psi(P)\\in\\mathscr{C}'_{XY}$ for all\n$\\varphi\\in\\cc_{\\mathrm{o}}(X^*)$, $\\psi\\in\\cc_{\\mathrm{o}}(Y^*)$, and $u\\in\\mathcal{H}_{X\/Y}$.  The\nclspan of the operators $a^*(u) \\psi(P)$ is $\\mathscr{T}_{XY}$, see\nProposition \\ref{pr:def3} and the comments after \\eqref{eq:L2a}, and\nfrom \\eqref{eq:cyzc} we have $\\mathscr{T}_X\\cdot\\mathscr{T}_{XY}=\\mathscr{T}_{XY}$.  Thus\nthe clspan of the operators $\\varphi(P) a^*(u) \\psi(P)$ is\n$\\mathscr{T}_{XY}$ for each $X\\supset Y$ and then we get\n$\\mathscr{T}_{XY}\\subset\\mathscr{C}'_{XY}$.  By taking adjoints we get\n$\\mathscr{T}_{XY}\\subset\\mathscr{C}'_{XY}$ if $X\\sim Y$.\n\nNow recall that the subspace $\\mathscr{T}^\\circ\\subset L(\\mathcal{H})$ defined by\n$\\mathscr{T}^\\circ_{XY}=\\mathscr{T}_{XY}$ if $X\\sim Y$ and $\\mathscr{T}^\\circ=\\{0\\}$ if\n$X\\not\\sim Y$ is a closed self-adjoint linear subspace of $\\mathscr{T}$ and\nthat $\\mathscr{T}^\\circ\\cdot\\mathscr{T}^\\circ=\\mathscr{C}$, cf. Theorem \\ref{th:tc}. By\nwhat we proved before we have $\\mathscr{T}^\\circ\\subset\\mathscr{C}'$ hence\n$\\mathscr{C}\\subset\\mathscr{C}'$. The converse inclusions is easy to prove. This\nfinishes the proof of Theorem \\ref{th:motiv}. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{S1}\n\\setcounter{equation}{0}\nThe Laguerre unitary ensemble ($ {\\rm LUE}_N $) refers to the eigenvalue probability\ndensity function (p.d.f.)\n\\begin{equation}\n \\frac{1}{C_{N,a}}\n \\prod^{N}_{l=1}\\lambda_l^ae^{-\\lambda_l}\\prod_{1 \\leq j<k \\leq N}(\\lambda_k-\\lambda_j)^2 , \n\\label{LUE_pdf}\n\\end{equation}\nwhere\n\\begin{equation}\n  C_{N,a} := \\prod^{N-1}_{j=0}\\Gamma(j+1)\\Gamma(j+a+1) ,\n\\label{LUE_norm}\n\\end{equation}\nwith support on $ \\lambda_l \\in [0,\\infty) $. For $ a \\in \\mathbb Z_{\\geq 0} $, this\neigenvalue p.d.f. is realised by non-negative matrices $ X^{\\dagger}X $ where \n$ X $ is an $ M\\times N $ complex Gaussian matrix and $ a=M-N $. In our work\n\\cite{FW_2002a} the average over (\\ref{LUE_pdf})\n\\begin{equation}\n  \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) := \\frac{C_{N,a}}{C_{N,a+\\mu}}\n  \\Big\\langle \\prod^N_{l=1}(1-\\xi\\chi^{(l)}_{(0,s)})(\\lambda_l-s)^{\\mu} \\Big\\rangle_{{\\rm LUE}_N} ,\n\\label{LUE_E}\n\\end{equation}\nwhere $ \\chi^{(l)}_{J}=1 $ for $ \\lambda_l \\in J $, $ \\chi^{(l)}_{J}=0 $ otherwise,\nand the normalisation is chosen so that \n\\begin{equation}\n  \\left.\\tilde{E}_{2,N}((0,s);a,\\mu;\\xi)\\right|_{s=0} := 1 ,\n\\label{LUE_Enorm}\n\\end{equation}\nwas characterised as a $ \\tau $-function for the Painlev\\'e V system. As a consequence,\nit was shown that \n\\begin{equation}\n W_N(s;a,\\mu;\\xi) \n  := s\\frac{d}{ds}\\log\\left( s^{-N\\mu}\\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) \\right) ,\n\\label{LUE_sigma}\n\\end{equation}\nsatisfies the Jimbo-Miwa-Okamoto $ \\sigma $-form of the Painlev\\'e V equation\n\\begin{equation}\n  (s\\sigma''_V)^2 - \\left[ \\sigma_V - s\\sigma'_V + 2(\\sigma'_V)^2 +\n  (\\sum^3_{j=0}\\nu_j) \\sigma'_V \\right]^2\n  + 4 \\prod^3_{j=0}(\\nu_j + \\sigma'_V) = 0 ,\n\\label{PV_sigma}\n\\end{equation}\nwith\n\\begin{equation}\n  \\nu_0 = 0, \\quad \\nu_1=-\\mu, \\quad \\nu_2 = N+a, \\quad \\nu_3 = N, \\quad\n  \\sum^3_{j=0}\\nu_j = 2N+a-\\mu .\n\\label{LUE_Vparam}\n\\end{equation}\n\nFor this to uniquely characterise $ W_N $, a boundary condition must be specified. \nHowever in \\cite{FW_2002a} only in the cases $ \\mu=0 $ and $ \\mu=2 $ were a\nboundary condition specified for general $ \\xi $.\n\nAlso considered in \\cite{FW_2002a} was the hard edge limiting average\n\\begin{equation}\n  \\tilde{E}^{\\rm hard}_N(t;a,\\mu;\\xi) :=\n  \\lim_{N \\to \\infty} \\Big(\n  \\frac{C_{N,a}}{C_{N,a+\\mu}}\\tilde{E}_{2,N}((0,\\frac{t}{4N});a,\\mu;\\xi) \\Big) .\n\\label{HE_Edefn}\n\\end{equation}\nIt was shown that\n\\begin{equation}\n  \\tilde{E}^{\\rm hard}_2(t;a,\\mu;\\xi) = \\exp\\int_{0}^t u^{h}(s;a,\\mu;\\xi)\\,\\frac{ds}{s} ,\n\\label{HE_E}\n\\end{equation}\nwhere setting\n\\begin{equation}\n  u^{h}(s;a,\\mu;\\xi) =-(\\sigma_{{\\rm III}'}(s)+\\frac{\\mu(\\mu+a)}{2}) ,\n\\label{HE_E.2}\n\\end{equation}\nthe function $ \\sigma_{{\\rm III}'}(s) $ satisfies the Jimbo-Miwa-Okamoto $ \\sigma $-form\nof the Painlev\\'e ${\\rm III^{\\prime}}\\;$ equation\n\\begin{equation}\n  (s\\sigma''_{{\\rm III}'})^2 - v_1v_2(\\sigma'_{{\\rm III}'})^2\n  + \\sigma'_{{\\rm III}'}(4\\sigma'_{{\\rm III}'}-1)(\\sigma_{{\\rm III}'}-s\\sigma'_{{\\rm III}'})\n  - \\frac{1}{4^3}(v_1-v_2)^2 = 0 ,\n\\label{PIII_sigma}\n\\end{equation}\nwith parameters\n\\begin{equation}\n  v_1 = a+\\mu, \\qquad v_2 = a-\\mu .\n\\label{HE_IIIparam}\n\\end{equation}\nAgain only in the cases $ \\mu=0 $ and $ \\mu=2 $ were boundary conditions specified for \ngeneral $ \\xi $.\n\nThe aim of this work is to specify the boundary conditions relevant to both \n(\\ref{LUE_sigma}) and (\\ref{PV_sigma}) for general values of the parameters. In the \ncase of (\\ref{LUE_sigma}) this is done by writing the average (\\ref{LUE_E}) in\nits equivalent determinant form and evaluating the matrix elements in terms of\ncertain $ {}_1F_1 $ hypergeometric functions. With the small $ s $ behaviour \nof the matrix elements determined, it turns out that the determinant is such that\nits corresponding small $ s $ behaviour can readily be deduced. The small $ s $\nasymptotic form of (\\ref{LUE_E}) then follows immediately. Scaling this \nasymptotic form as required by (\\ref{HE_Edefn}) then gives the small $ t $\nbehaviour of $ \\tilde{E}^{\\rm hard}_N(t;a,\\mu;\\xi) $ and the small $ s $ behaviour\nof $ u^{h}(s;a,\\mu;\\xi) $.\n\nThe general small $ s $ asymptotic form of the permitted solutions of \n(\\ref{PV_sigma}) and (\\ref{PIII_sigma}) have been given by Jimbo \\cite{Ji_1982}.\nAs part of this study the boundary conditions found here are compared against\nthese general forms. It is found that in both cases only one of the two branches\npermitted by the general solution is present in our random matrix problem.\n\nAs an application of our results we specify the rapid computation of the power\nseries expansion of a certain Hankel determinant of Bessel functions. The latter\nis known form our work \\cite{FW_2002a} to be a special case of \n$ \\tilde{E}^{\\rm hard}_N(t;a,\\mu;\\xi) $. The coefficients in the power series appear\nin an asymptotic formula obtained recently by Conrey, Rubinstein and Snaith \n\\cite{CRS_2005} for the integer moments of the derivative of the characteristic\npolynomial of a unitary random matrix. This in turn has application to the study\nof the derivative of the Riemann zeta function on the critical line.\n \n\\section{Small $ s $ Expansion of $ \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) $}\\label{S2}\n\\setcounter{equation}{0}\nA standard result in random matrix theory, which in fact goes back to an identity\nof Heine (see \\cite{ops_Sz}) expresses the random matrix average (\\ref{LUE_E}) as\na determinant\n\\begin{equation}\n \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) \n  = \\frac{N!C_{N,a}}{C_{N,a+\\mu}}\\det[w_{j+k}]_{j,k=0,\\ldots,N-1} , \n\\label{LUE_Hdet}\n\\end{equation}\nwhere\n\\begin{equation}\n  w_{n} \n  := \\int^{\\infty}_0 d\\lambda\\; (1-\\xi\\chi_{(0,s)})(\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda} .\n\\label{LUE_Hsym}\n\\end{equation}\nUnless $ \\mu $ is a non-negative integer (\\ref{LUE_Hsym}) is not well defined for\n$ s $ real and positive, which is the domain of interest. To remedy this, we note \nthat simple manipulation gives\n\\begin{equation}\n  w_{n} \n  := \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n     + (1-\\xi)\\int^{s}_0 d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda} ,\n\\label{LUE_Hsym.2}\n\\end{equation}\nand in the second integral of this expression write \n$ (\\lambda-s)^{\\mu} = e^{\\mu\\log(\\lambda-s)} $ with \n$ -\\pi < \\arg\\log(\\lambda-s) \\leq \\pi $. We then obtain\n\\begin{equation}\n  w_{n} \n  := \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n     + (1-\\xi)e^{\\pi i\\mu}\\int^{s}_0 d\\lambda\\; (s-\\lambda)^{\\mu}\\lambda^{n+a}e^{-\\lambda} ,\n\\label{LUE_Hsym.3}\n\\end{equation}\nwhich is well defined for $ \\Re(\\mu) > -1 $ and $ \\Re(a) > -1 $ for \n$ s>0 $ with the additional constraint $ \\Re(\\mu+a) > -1 $ at $ s=0 $.\n\nWe seek the leading terms in the small $ s$ expansion of (\\ref{LUE_Hsym.3}). \nThese can be read off from an explicit evaluation in terms of the $ {}_1F_1 $\nconfluent hypergeometric function \\cite{WW_1965}. \n\\begin{proposition}\nSubject to the conditions $ \\Re(\\mu) > -1 $, $ \\Re(a) > -1 $,\n$ \\Re(\\mu+a) > -1 $ and $ \\mu+a\\notin \\mathbb Z_{\\geq 0} $ we have\n\\begin{equation}\n  w_{n} = a_n(s) + s^{n+\\mu+a+1}b_n(s) ,\n\\label{LUE_Hsym.4}\n\\end{equation}\nwhere $ a_n(s), b_n(s) $ are analytic about $ s=0 $ and given explicitly by\n\\begin{equation}\n\\begin{split}\n  a_n(s) &=\n  \\Gamma(\\mu+n+a+1)e^{-s}{}_1F_1(-a-n;-\\mu-a-n;s) ,\n  \\\\\n  b_n(s) &=\n  \\frac{\\Gamma(\\mu+1)\\Gamma(n+a+1)}{\\Gamma(\\mu+n+a+2)}\n  \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right)\n  \\\\\n         & \\qquad\\times\n  e^{-s}{}_1F_1(\\mu+1;\\mu+a+n+2;s) .\n\\end{split}\n\\label{Hsym_1F1}\n\\end{equation}\nIn particular, under the above conditions,\n\\begin{equation}\n  w_{n} \\mathop{\\sim}\\limits_{s \\to 0} a_n(0)+sa'_n(0)+s^{n+\\mu+a+1}b_n(0) ,\n\\label{LUE_Hsym.5}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n  a_n(0)  &= \\Gamma(\\mu+n+a+1) ,\n  \\\\\n  a'_n(0) &= -\\mu\\Gamma(\\mu+n+a) ,\n  \\\\\n  b_n(0) &=\n  \\frac{\\Gamma(\\mu+1)\\Gamma(n+a+1)}{\\Gamma(\\mu+n+a+2)}\n  \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) .\n\\end{split}\n\\label{Hsym_1F1_exp}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe results (\\ref{LUE_Hsym.5}) and (\\ref{Hsym_1F1_exp}) are immediate corollaries\nof (\\ref{LUE_Hsym.4}) and (\\ref{Hsym_1F1}) and the fact that\n\\begin{equation*}\n  {}_1F_1(\\gamma;\\alpha;s) = 1+\\frac{\\gamma}{\\alpha}s+{\\rm O}(s^2) .\n\\end{equation*}\nTo derive (\\ref{LUE_Hsym.4}), we note that simple manipulation shows\n\\begin{equation*}\n  \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n = s^{a+n}e^{-s}\\int^{\\infty}_0 d\\lambda\\; (1+\\lambda\/s)^{n+a}\\lambda^{\\mu}e^{-\\lambda} .\n\\end{equation*}\nBut with \n\\begin{equation*}\n  W_{k,m}(z) = \\frac{z^ke^{-z\/2}}{\\Gamma(1\/2-k+m)}\n   \\int^{\\infty}_0 dt\\; (1+t\/z)^{k-1\/2+m}t^{-k-1\/2+m}e^{-t} ,\n\\end{equation*}\nspecifying the Whittaker function, it is known that \\cite{WW_1965}\n\\begin{equation*}\n  W_{k,m}(z)\n  = \\frac{\\Gamma(-2m)}{\\Gamma(1\/2-k-m)}M_{k,m}(z)+\\frac{\\Gamma(2m)}{\\Gamma(1\/2-k+m)}M_{k,-m}(z)\n\\end{equation*}\nwhere\n\\begin{equation*}\n  M_{k,m}(z) = z^{m+1\/2}e^{-z\/2}{}_1F_1(1\/2-k+m;2m+1;z) .\n\\end{equation*}\nConsequently\n\\begin{multline}\n  \\int^{\\infty}_s d\\lambda\\; (\\lambda-s)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n =  \\Gamma(\\mu+a+n+1)e^{-s}{}_1F_1(-a-n;-\\mu-a-n;s) \n \\\\\n  + \\frac{\\Gamma(\\mu+1)\\Gamma(-\\mu-a-n-1)}{\\Gamma(-a-n)}\n    s^{\\mu+a+n+1}e^{-s}{}_1F_1(\\mu+1;\\mu+a+n+2;s) .\n\\label{ws6}\n\\end{multline}\nThe left-hand side of (\\ref{ws6}) exists for $ \\Re(\\mu)>-1 $ if $ s>0 $ and\n$ \\Re(\\mu+a)>-1 $ if $ s=0 $, whereas the right-hand side is valid in \nthis parameter domain except for $ \\mu+a+n\\in\\mathbb Z_{\\geq 0} $, and in this case the individual\nterms have a simple pole at $ a+n\\notin\\mathbb Z_{\\geq 0} $ or are undefined when $ a+n\\in\\mathbb Z_{\\geq 0} $. \nNeedless to say the sum of the terms on the right-hand side has the same analytic\ncharacter as the left-hand side.\n\nRegarding the second integral in (\\ref{LUE_Hsym.3}), we first note that a simple \nchange of variables gives\n\\begin{equation*}\n  \\int^{s}_0 d\\lambda\\; (s-\\lambda)^{\\mu}\\lambda^{n+a}e^{-\\lambda}\n  = s^{n+1+a+\\mu}e^{-s}\\int^{1}_0 dx\\; (1-x)^{n+a}x^{\\mu}e^{sx} .\n\\end{equation*}\nBut the integral on the right hand side is the Euler integral representation of\nthe $ {}_1F_1 $ function, which shows\n\\begin{multline}\n \\int^{s}_0 d\\lambda\\; (s-\\lambda)^{\\mu}\\lambda^{n+a}e^{-\\lambda} \n \\\\\n  = \\frac{\\Gamma(\\mu+1)\\Gamma(a+n+1)}{\\Gamma(\\mu+a+n+2)}\n    s^{\\mu+a+n+1}e^{-s}{}_1F_1(\\mu+1;\\mu+a+n+2;s) .\n\\label{ws7}\n\\end{multline}\nThis latter relation is valid for $ \\Re(\\mu)>-1 $ and $ \\Re(a)>-1 $ when\n$ s>0 $.\nSubstituting (\\ref{ws6}) and (\\ref{ws7}) in (\\ref{LUE_Hsym.3}) and using the \nappropriate gamma function identities gives (\\ref{LUE_Hsym.4}), (\\ref{Hsym_1F1}).\n\\end{proof}\n\nWhen $ \\mu+a \\in\\mathbb Z_{\\geq 0} $ we have to consider two exceptional cases where one of the \nhypergeometric functions are not defined - the first when $ a+n \\in\\mathbb Z_{\\geq 0} $ \nfor which the hypergeometric function is indeterminate, and the second when\n$ a+n \\notin\\mathbb Z_{\\geq 0} $ and the hypergeometric function has a simple pole.\nThese two cases can be recovered by taking suitable limits and we just state the\nfinal results.\n\n\\begin{proposition}\nWhen $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and $ a+n=k \\in\\mathbb Z_{\\geq 0} $ with $ n+j\\geq k $ \nwe have\n\\begin{multline}\n w_n = k!e^{-s}\\Bigg\\{  \\sum^k_{l=0}\\frac{(n+j-l)!}{(k-l)!l!}s^l\n \\\\\n  + (-1)^{n+j+k}(1-\\xi)\\frac{(n+j-k)!}{(n+j+1)!}s^{n+j+1}\n    {}_1F_1(n+j+1-k;n+j+2;s) \\Bigg\\} ,\n\\label{Hsym_indeterm}\n\\end{multline}\nand to leading order in small $ s $ we have\n\\begin{equation}\n  w_{n} \\mathop{\\sim}\\limits_{s \\to 0} \n  (n+j)! - (n+j-k)(n+j-1)!s + (-1)^{n+j+k}(1-\\xi)\\frac{(n+j-k)!k!}{(n+j+1)!}s^{n+j+1} .\n\\label{Exp_indeterm}\n\\end{equation}\n\\end{proposition}\nNote that the condition $ n+j\\geq k $ is the same as $ \\mu \\geq 0 $, which falls\nwithin the domain of interest. The key difference of (\\ref{Exp_indeterm}) with\n(\\ref{LUE_Hsym.5}) and (\\ref{Hsym_1F1_exp}) is that the non-analytic term is\nnow polynomial and the second part of this term is absent having been cancelled by\na counterbalancing term.\n\n\\begin{proposition}\nWhen $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and $ a+n \\notin\\mathbb Z_{\\geq 0} $ we have \n\\begin{multline}\n w_n = e^{-s}\\Bigg\\{ \\sum^{n+j}_{l=0}\\frac{(-a-n)_l(n+j-l)!}{l!}(-s)^l \n \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+n+1)}{(n+j+1)!}(1-\\xi)e^{i\\pi\\mu}s^{n+j+1}\n   {}_1F_1(\\mu+1;n+j+2;s)\n \\\\\n  + (-1)^j\\frac{\\sin\\pi a}{\\pi}\\frac{\\Gamma(\\mu+1)\\Gamma(a+n+1)}{(n+j+1)!}s^{n+j+1}\n \\\\\n \\times\n    \\sum^{\\infty}_{l=0}\\left[\\psi(l+1)+\\psi(n+j+l+2)-\\psi(\\mu+l+1)-\\log s \\right]\n          \\frac{(\\mu+1)_l}{(n+j+2)_l}\\frac{s^l}{l!} \\Bigg\\} ,\n\\label{Hsym_pole}\n\\end{multline}\nand its leading order behaviour for small $ s $ is\n\\begin{multline}\n  w_{n} \\mathop{\\sim}\\limits_{s \\to 0} \n  (n+j)!+(a-j)(n+j-1)!s \n  \\\\\n  +\\frac{(a-j)_{n+j+1}}{(n+j+1)!}s^{n+j+1}\n   \\left[\\frac{\\pi e^{-i\\pi a}}{\\sin\\pi a}(1-\\xi)+\\psi(1)+\\psi(n+j+2)-\\psi(\\mu+1)-\\log s \\right] .\n\\label{Exp_pole}\n\\end{multline}\n\\end{proposition}\nThe expansion (\\ref{Exp_pole}) differs significantly from (\\ref{LUE_Hsym.5}) and \n(\\ref{Hsym_1F1_exp}) because of the presence of logarithmic terms which now \nreplace the non-analytic contributions of the generic case.\n\n\\begin{corollary}\nUnder generic conditions on $ \\mu+a $ we have\n\\begin{multline}\n  \\det[w_{j+k}]_{j,k=0,\\ldots,N-1} \n  \\\\\n  = \\det[\\Gamma(\\mu+a+1+j+k)]_{j,k=0,\\ldots,N-1}\n  \\\\\n    - \\mu s\\det[\\Gamma(\\mu+a+j) \\;\\; \\Gamma(\\mu+a+1+j+k)]_{{j=0,\\ldots,N-1}\\atop\n                                                         {k=1,\\ldots,N-1}}\n    + {\\rm O}(s^2)\n  \\\\\n    + s^{\\mu+a+1}b_0(0)\\det[\\Gamma(\\mu+a+3+j+k)]_{j,k=0,\\ldots,N-2}\n      \\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\\n    + {\\rm O}(s^{2(\\mu+a+1)}) .\n\\label{Hdet_exp}\n\\end{multline}\n\\end{corollary}\n\\begin{proof}\nAccording to (\\ref{LUE_Hsym.5})\n\\begin{multline*}\n  \\det[w_{j+k}]_{j,k=0,\\ldots,N-1} \n  \\\\\n  \\mathop{\\sim}\\limits_{s \\to 0}\n  \\det[a_{j+k}(0)+sa'_{j+k}(0)+s^{\\mu+a+1+j+k}b_{j+k}(0)]_{j,k=0,\\ldots,N-1}\n  \\\\\n  \\mathop{\\sim}\\limits_{s \\to 0}\n  \\det[a_{j+k}(0)]_{j,k=0,\\ldots,N-1}+s[s]\\det[a_{j+k}(0)+sa'_{j+k}(0)]_{j,k=0,\\ldots,N-1}\n  \\\\\n  + s^{\\mu+a+1}b_0(0)\\det[a_{j+k+2}(0)]_{j,k=0,\\ldots,N-2} ,\n\\end{multline*}\nwhere $ [s]P(s) $ denotes the coefficient of $ s $ in $ P(s) $. Recalling the \nexplicit formula for $ a_n(0) $ as given in (\\ref{Hsym_1F1_exp}) we obtain the \nconstant term and the term proportional to $ s^{\\mu+a+1} $ in (\\ref{Hdet_exp}).\nIt remains to compute the coefficient of $ s $, which according to \n(\\ref{Hsym_1F1_exp}) has the explicit form\n\\begin{equation}\n   [s]\\det[\\Gamma(\\mu+a+1+j+k)-\\mu s\\Gamma(\\mu+a+j+k)]_{j,k=0,\\ldots,N-1} .\n\\label{ws9}\n\\end{equation}\nUsing the linearity formula\n\\begin{equation*}\n   \\det[{\\bf a}_1 \\cdots {\\bf a}_j+{\\bf b}_j \\cdots {\\bf a}_n]\n   = \\det[{\\bf a}_1 \\cdots {\\bf a}_j \\cdots {\\bf a}_n]\n    +\\det[{\\bf a}_1 \\cdots {\\bf b}_j \\cdots {\\bf a}_n] ,\n\\end{equation*}\nwhere the $ \\bf{a} $'s and $ \\bf{b} $'s are column vectors, on each column of\nthe determinant we see that of the terms proportional to $ s $ only the one\nobtained from expanding the first column in non-zero (all the rest result in \ntwo identical columns), and the determinant given by (\\ref{Hdet_exp}) results.\n\\end{proof}\n\nIt remains to evaluate the determinants. For this task we make use of the identity\n\\cite{Nd_2004}\n\\begin{equation*}\n   \\det[\\Gamma(z_k+j)]_{j,k=0,\\ldots,n-1}\n   = \\prod^{n-1}_{j=0}\\Gamma(z_j)\\prod_{0\\leq j<k\\leq n-1}(z_k-z_j) .\n\\end{equation*}\nAfter straightforward manipulations, gamma function evaluations of all the \ndeterminants in (\\ref{Hdet_exp}) can be obtained. Substituting in (\\ref{LUE_Hdet}),\nand recalling that the normalisation is such that at $ s=0 $ $ \\tilde{E}_N $ \nis equal to unity, we obtain the sought small $ s $ expansion of $ \\tilde{E}_N $\nand thus $ W_N $ valid for general values of the parameters.\n\n\\begin{proposition}\\label{LUE_exp}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ and $ \\mu+a\\notin\\mathbb Z_{\\geq 0} $ we have \n\\begin{multline}\n  \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) = 1-\\frac{\\mu N}{\\mu+a}s+{\\rm O}(s^2)\n  \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)\\Gamma(\\mu+a+N+1)}{\\Gamma^2(\\mu+a+2)\\Gamma(\\mu+a+1)\\Gamma(N)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   s^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) ,\n\\label{LUE_Eexp}\n\\end{multline}\nand consequently\n\\begin{multline}\n  W_{N}(s;a,\\mu;\\xi) = -N\\mu-\\frac{\\mu N}{\\mu+a}s+{\\rm O}(s^2)\n  \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)\\Gamma(\\mu+a+N+1)}{\\Gamma(\\mu+a+2)\\Gamma^2(\\mu+a+1)\\Gamma(N)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   s^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) .\n\\label{LUE_Wexp}\n\\end{multline}\n\\end{proposition}\nIn the first exceptional case $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and $ a=k \\in\\mathbb Z_{\\geq 0} $ \nwith $ j\\geq k $ one can still use (\\ref{LUE_Eexp}) but omitting the term involving\nthe ratio of sines, in the case $ j=0 $, or the whole term if $ j>0 $.\nThe situation of the other exceptional case $ \\mu+a=j \\in\\mathbb Z_{\\geq 0} $ and \n$ a \\notin\\mathbb Z_{\\geq 0} $ is more complicated and more so for larger $ j $, and we \nonly give the examples of $ j=0,1 $.\n\\begin{proposition}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ with $ \\mu+a=0 $ we have \n\\begin{multline}\n\\tilde{E}_{2,N}((0,s);a,\\mu=-a;\\xi) = 1\n  \\\\\n  +\\Big\\{-1 + \\frac{\\pi a}{\\sin\\pi a}e^{-i\\pi a}(1-\\xi)\n   +a\\left[2\\psi(2)+\\psi(1)-\\psi(1-a)-\\psi(N+1)-\\log s \\right] \\Big\\}Ns\n  \\\\ + {\\rm o}(s) .\n\\label{LUE_Eexp_pole0}\n\\end{multline}\nFor $ \\mu+a=1 $ we have\n\\begin{multline}\n\\tilde{E}_{2,N}((0,s);a,\\mu=1-a;\\xi) = 1+(a-1)Ns\n  \\\\\n  +\\frac{a(a-1)}{4}\\Big\\{\\frac{\\pi}{\\sin\\pi a}e^{-i\\pi a}(1-\\xi)\n   +2\\psi(3)+\\psi(2)-\\psi(2-a)-\\psi(N+2)-\\log s \\Big\\}(N+1)Ns^2\n  \\\\ + {\\rm o}(s^2) .\n\\label{LUE_Eexp_pole1}\n\\end{multline}\n\\end{proposition}\n\n\\section{Comparison with the Jimbo solution}\\label{S3}\n\\setcounter{equation}{0}\nThe small $ s$ expansion of the most general solution permitted by (\\ref{PV_sigma}),\nor more precisely its corresponding $ \\tau $-function (see (\\ref{PV_tau}) below) has \nbeen determined by Jimbo \\cite{Ji_1982}. However in \\cite{Ji_1982} the equation\n(\\ref{PV_sigma}) is not treated directly. Instead the discussion is based on\nthe equation\n\\begin{multline}\n  (s\\zeta'')^2 - \\left[ \\zeta - s\\zeta' + 2(\\zeta')^2 -\n  (2\\theta_0+\\theta_{\\infty}) \\zeta' \\right]^2 \\\\\n  + 4\\zeta'(\\zeta'-\\theta_0)(\\zeta'-\\frac{1}{2}(\\theta_0-\\theta_s+\\theta_{\\infty}))\n    (\\zeta'-\\frac{1}{2}(\\theta_0+\\theta_s+\\theta_{\\infty}))\n  = 0 ,\n\\label{PV_zeta}\n\\end{multline}\nand the small $ s $ behaviour of the corresponding $ \\tau $-function $ \\tau_V(s) $,\nspecified by the the requirement that\n\\begin{equation}\n  \\zeta(s) = s\\frac{d}{ds}\\log\\tau_V(s)+\\frac{1}{2}(\\theta_0+\\theta_{\\infty})s\n          +\\frac{1}{4}[(\\theta_0+\\theta_{\\infty})^2-\\theta_s^2] , \n\\label{PV_tau}\n\\end{equation}\nwas determined.\n\nComparison of (\\ref{PV_zeta}), (\\ref{PV_tau}) with (\\ref{PV_sigma}), \n(\\ref{LUE_sigma}) shows that for the parameters (\\ref{LUE_Vparam})\n\\begin{equation}\n  \\tilde{E}_{2,N}((0,s);a,\\mu;\\xi) = s^{N^2+N(a+\\mu)}e^{-(N+a\/2)s}\\tau_V(s) ,\n\\label{LUE_Vtau}\n\\end{equation}\nwhile in general\n\\begin{equation}\n  \\theta_0 = -\\nu_1, \\quad \\theta_s = \\nu_2-\\nu_3, \n  \\quad \\theta_{\\infty} = \\nu_1-\\nu_2-\\nu_3 .\n\\label{Vnu_theta}\n\\end{equation}\nNote that for the parameters (\\ref{LUE_Vparam}) we thus thus have\n\\begin{equation}\n  \\theta_0 = \\mu, \\quad \\theta_s = a, \\quad \\theta_{\\infty} = -2N-a-\\mu .\n\\label{LUE_theta}\n\\end{equation}\n\nThe relevant result from \\cite{Ji_1982} can now be presented. It states that the\nmost general small $ s $ behaviour of $ \\tau_V(s) $ permitted by the equation\n(\\ref{PV_zeta}) is\n\\begin{multline}\n  \\tau_V(s) = Cs^{(\\sigma^2-\\theta^2_{\\infty})\/4}\n       \\Bigg\\{ 1\n               - \\frac{\\theta_{\\infty}(\\theta^2_s-\\theta^2_0+\\sigma^2)}\n                     {4\\sigma^2}s \\\\\n   + u\n      \\frac{\\Gamma^2(-\\sigma)}{\\Gamma^2(2+\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle\\theta_s+\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_s-\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_{\\infty}+\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle\\theta_s+\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_s-\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_{\\infty}-\\sigma}{\\displaystyle 2})}\n      s^{1+\\sigma}\n   \\\\\n   + \\frac{1}{u}\n      \\frac{\\Gamma^2(\\sigma)}{\\Gamma^2(2-\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle\\theta_s+\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_s-\\theta_0-\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle\\theta_{\\infty}-\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle\\theta_s+\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_s-\\theta_0+\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle\\theta_{\\infty}+\\sigma}{\\displaystyle 2})}\n      s^{1-\\sigma}\n   \\\\\n   + {\\rm O}(|s|^{2(1-\\Re(\\sigma))}) \\Bigg\\} ,\n\\label{PV_sExp_0}\n\\end{multline}\nwhere $ C $ is a normalisation constant, while $ u $ and $ \\sigma $ are arbitrary\nparameters. The above result was derived subject to the conditions \n$ \\theta_0, \\theta_s \\notin\\mathbb Z $,\n$ \\frac{1}{2}(\\theta_{\\infty}\\pm\\sigma) \\notin\\mathbb Z $,\n$ \\frac{1}{2}(\\theta_{s}\\pm\\theta_0\\pm\\sigma) \\notin\\mathbb Z $ and\nthat $ 0 < \\Re(\\sigma) < 1 $ (a distinct solution was presented for $ \\sigma=0 $).\nThese conditions therefore strictly apply only to the generic or transcendental solutions of\nthe fifth Painlev\\'e equation. \nFor generic parameter values the terms given in (\\ref{PV_sExp_0}) uniquely \nspecify all the subsequent terms in the convergent Puisuex-type expansion for \n$ \\zeta(s) $ about $ s=0 $\n\\begin{equation}\n   \\zeta(s) = \n   \\sum^{\\infty}_{j=0}\\sum_{|k|\\leq j}c_{j,k}s^{j+k\\sigma} ,\n\\label{PV_puisuex}\n\\end{equation}\ni.e. with any two of $ c_{1,0},c_{1,1} $ or $ c_{1,-1} $ given. \n\nTo relate this to $ \\tilde{E}_{2,N} $, we see from (\\ref{LUE_Vtau}) and \n(\\ref{LUE_theta}) that we require $ \\sigma^2=(a+\\mu)^2 $ and thus we can choose\n\\begin{equation}\n  \\sigma=a+\\mu .\n\\label{LUE_const}\n\\end{equation}\nThis relation, $ \\sigma=\\theta_0+\\theta_{s} $, is a violation of one of the strict \nconditions given above and is in fact a sufficient condition for a classical solution,\nalong with the necessary condition $ \\theta_0+\\theta_s+\\theta_{\\infty}=-2N\\in\\mathbb Z $,\nwhich is the type of solution that we are dealing with here. \nHowever we conjecture that Jimbo's conditions\ncan be relaxed to accommodate such solutions and the corresponding formulae \n(or limiting forms if necessary) still hold. \nWith this choice of $ \\sigma $ the coefficient of $ s^{1-\\sigma} $ in (\\ref{PV_sExp_0})\ncontains a factor of\n\\begin{equation*}\n  \\frac{1}{\\Gamma(\\frac{\\displaystyle\\theta_{\\infty}+\\sigma}{\\displaystyle 2})}\n  = \\frac{1}{\\Gamma(-N)}\n\\end{equation*}\nand thus vanishes. Simplifying the other terms gives\n\\begin{multline*}\n  \\tau_V(s) \\sim Cs^{-N^2-N(a+\\mu)}\n       \\Bigg\\{ 1 + \\frac{(2N+a+\\mu)a}{2(a+\\mu)}s \\\\\n   + u \\frac{\\sin\\pi\\mu}{\\sin\\pi(a+\\mu)}\n       \\frac{\\Gamma(a+1)\\Gamma(\\mu+1)\\Gamma(N+1+a+\\mu)}\n            {\\Gamma^2(2+a+\\mu)\\Gamma(1+a+\\mu)\\Gamma(N)}\n      s^{1+a+\\mu} \\Bigg\\} .\n\\end{multline*}\nSubstituting in (\\ref{LUE_Vtau}) we see that this is in precise agreement with \n(\\ref{LUE_Eexp}) provided we choose\n\\begin{equation}\n  u\\frac{\\sin\\pi\\mu}{\\sin\\pi(a+\\mu)} = (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(a+\\mu)}\n\\label{LUE_u}\n\\end{equation}\n\n\\section{The hard edge limit}\\label{S4}\n\\setcounter{equation}{0}\nThe hard edge limit is defined by (\\ref{HE_Edefn}). However, only in the cases\n$ \\mu=0 $, $ \\mu=2 $ do we know how to prove its existence for general $ \\xi $\n(in the case $ \\mu=0 $ $ \\tilde{E}_{2,N} $ can be written as a Fredholm determinant,\nwhile the case $ \\mu=2 $ is related to this via differentiation). However a log-gas\nviewpoint (\\cite{rmt_Fo}) indicates that the limit will be well defined, and\nmoreover we expect that it can be taken term-by-term in the small $ s $ expansion\nof $ \\tilde{E}_{2,N} $. In this section we will show that taking the hard edge \nlimit of the small $ s $ expansion (\\ref{LUE_Eexp}) give rise to an initial \ncondition for the solution of (\\ref{PIII_sigma}) consistent with that allowed\nby Jimbo's theory of the small $ s $ expansion of the Painlev\\'e ${\\rm III^{\\prime}}\\;$ equation. From a \npractical perspective this specifies $ \\tilde{E}^{\\rm hard}_{2} $ for general\nvalues of the parameters according to (\\ref{HE_E}), while from a theoretical\nviewpoint it lends weight to the belief that (\\ref{HE_E}) is indeed the correct\nlimiting evaluation for general values of the parameters.\n\nUnder the assumption that the hard edge limit can be taken term-by-term in the\nsmall $ s $ expansion of Proposition (\\ref{LUE_exp}) is immediate.\n\n\\begin{corollary}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ and $ \\mu+a\\notin\\mathbb Z_{\\geq 0} $ \nwe have\n\\begin{multline}\n  \\tilde{E}^{\\rm hard}_{2}(s;a,\\mu;\\xi) = 1-\\frac{\\mu}{4(a+\\mu)}s+{\\rm O}(s^2)\n  \\\\\n  +\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)}{\\Gamma^2(\\mu+a+2)\\Gamma(\\mu+a+1)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   \\left(\\frac{s}{4}\\right)^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) ,\n\\label{HE_Eexp}\n\\end{multline}\nand consequently the $ \\sigma $-function $ \\sigma_{\\rm III'}(s) $ in (\\ref{HE_E.2})\nhas the small $ s $ expansion\n\\begin{multline}\n  \\sigma_{\\rm III'}(s) = -\\frac{\\mu(\\mu+a)}{2}+\\frac{\\mu}{4(\\mu+a)}s+{\\rm O}(s^2)\n  \\\\\n  -\\frac{\\Gamma(\\mu+1)\\Gamma(a+1)}{\\Gamma(\\mu+a+2)\\Gamma^2(\\mu+a+1)}\n   \\left( (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(\\mu+a)} \\right) \n   \\left(\\frac{s}{4}\\right)^{\\mu+a+1}\\left\\{1+{\\rm O}(s)\\right\\}\n  \\\\ +{\\rm O}(s^{2(\\mu+a+1)}) .\n\\label{HE_Sexp}\n\\end{multline}\n\\end{corollary}\n\nSome examples of exceptional cases not covered by the preceding corollary\nare the following. They are obtained by taking the hard edge limit of\n(\\ref{LUE_Eexp_pole0}) and (\\ref{LUE_Eexp_pole1}). \n\\begin{corollary}\nFor $ \\Re(\\mu)>-1 $, $ \\Re(a)>-1 $ and $ \\mu+a=0 $ we have \n\\begin{multline}\n\\tilde{E}^{\\rm hard}_{2}(s;a,\\mu=-a;\\xi) = 1\n  \\\\\n  +\\Big\\{\n   -1+\\frac{\\pi a}{\\sin\\pi a}e^{-\\pi ia}(1-\\xi)\n     +a[2\\psi(2)+\\psi(1)-\\psi(1-a)-\\log(s\/4)] \\Big\\}\\frac{s}{4}\n  \\\\ +{\\rm o}(s) ,\n\\label{HE_Eexp0}\n\\end{multline}\nwhilst for $ \\mu+a=1 $ we have\n\\begin{multline}\n\\tilde{E}^{\\rm hard}_{2}(s;a,\\mu=1-a;\\xi) = 1+(a-1)\\frac{s}{4}\n  \\\\\n  +\\frac{a(a-1)}{4} \\Big\\{\n   \\frac{\\pi}{\\sin\\pi a}e^{-\\pi ia}(1-\\xi)\n     +2\\psi(3)+\\psi(2)-\\psi(2-a)-\\log(s\/4) \\Big\\}\\left(\\frac{s}{4}\\right)^2\n  \\\\ +{\\rm o}(s^2) .\n\\label{HE_Eexp1}\n\\end{multline}\n\\end{corollary}\n\nTo compare these results to the small independent variable expansions given by \nJimbo in the theory of ${\\rm III^{\\prime}}\\;$, we must first undertake some preliminary\ncalculations as the equation (\\ref{PIII_sigma}) is not directly studied in \n\\cite{Ji_1982}. Rather the equation studied is\n\\begin{equation}\n  (t\\zeta'')^2 = 4\\zeta'(\\zeta' -1)(\\zeta - t\\zeta')\n  + \\left( \\frac{v_1+v_2}{2}-v_1\\zeta'\\right)^2 ,\n\\label{PIII_zeta}\n\\end{equation}\nwhere we have identified $ \\theta_0=-v_2 $, $ \\theta_{\\infty}=-v_1 $ \n($ \\theta_0, \\theta_{\\infty} $ are the parameters appearing in \\cite{Ji_1982}).\nIn terms of $ \\zeta(t) $ the $ \\tau $-function $ \\tau_{\\rm III'}(t) $ is specified\nby the requirement that\n\\begin{equation}\n  \\zeta(t) = t\\frac{d}{dt}\\log\\tau_{\\rm III'}(t)+\\frac{v_2^2-v_1^2}{4}+t ,\n\\label{PIII_zeta_tau}\n\\end{equation}\nand it is the small $ t $ expansion of $ \\tau_{\\rm III'}(t) $ presented in \n\\cite{Ji_1982}. Comparison of (\\ref{PIII_zeta}) and (\\ref{PIII_sigma}) shows that\n\\begin{equation}\n  \\zeta(t) = -\\sigma_{\\rm III'}(s)+\\frac{v_1(v_2-v_1)}{4}+\\frac{s}{4} ,\n  \\qquad t = \\frac{s}{4} .\n\\label{HE_zeta}\n\\end{equation}\nRecalling (\\ref{HE_E.2}), (\\ref{HE_E}), (\\ref{HE_zeta}) and (\\ref{PIII_zeta_tau}) \nwe see\n\\begin{equation}\n \\tilde{E}^{\\rm hard}_{2}(s;a,\\mu;\\xi) = t^{(v_2^2-v_1^2)\/4}\\tau_{\\rm III'}(t) .\n\\label{HE_tau}\n\\end{equation}\n\nIn \\cite{Ji_1982} the most general small $ t $ expansion of $ \\tau_{\\rm III'}(t) $\nas permitted by (\\ref{PIII_zeta}) is presented. It reads\n\\begin{multline}\n  \\tau_{\\rm III'}(t) = Ct^{(\\sigma^2-v^2_2)\/4}\n       \\Bigg\\{ 1 + \\frac{v_1v_2-\\sigma^2}{2\\sigma^2}t\n   \\\\\n   - u\n      \\frac{\\Gamma^2(-\\sigma)}{\\Gamma^2(2+\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle v_2+\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle v_1+\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle v_2-\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n      t^{1+\\sigma}\n   \\\\\n   - \\frac{1}{u}\n      \\frac{\\Gamma^2(\\sigma)}{\\Gamma^2(2-\\sigma)}\n      \\frac{\\Gamma(1+\\frac{\\displaystyle v_2-\\sigma}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n           {\\Gamma(\\frac{\\displaystyle v_2+\\sigma}{\\displaystyle 2})\n            \\Gamma(\\frac{\\displaystyle v_1+\\sigma}{\\displaystyle 2})}\n      t^{1-\\sigma}\n   \\\\\n   + {\\rm O}(|t|^{2(1-\\Re(\\sigma))}) \\Bigg\\} ,\n\\label{PIII_tExp_0}\n\\end{multline}\nwhere as in (\\ref{PV_sExp_0}) $ C $ is a normalisation, while $ u $ and $ \\sigma $\nare arbitrary parameters. This result was established under the assumptions that\n$ \\frac{1}{2}(v_1\\pm\\sigma) \\notin\\mathbb Z $ and $ \\frac{1}{2}(v_2\\pm\\sigma) \\notin\\mathbb Z $\nalong with $ 0 < \\Re\\sigma < 1 $ (for $ \\sigma=0 $ a distinct solution is given).\n\nTo see that this structure is consistent with (\\ref{HE_Eexp}) and (\\ref{HE_tau}),\nrecalling (\\ref{HE_IIIparam}) we see that for the right hand side of \n(\\ref{HE_tau}) to tend to $ 1 $ as $ t $ tends to zero we must have $ C=1 $ and\n$ \\sigma=\\pm v_1 $. Again this is a violation of first condition given above but we\nconjecture that the formulae have meaning under the following limiting procedure and\nare correct. Choosing the positive sign for definiteness, and then writing \n\\begin{equation*}\n   \\frac{u}{\\Gamma(\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n  = \\frac{u(v_1-\\sigma)}{2\\Gamma(1+\\frac{\\displaystyle v_1-\\sigma}{\\displaystyle 2})}\n\\end{equation*}\nwe see that requiring \n\\begin{equation*}\n  \\frac{u}{2}(v_1-\\sigma) \\to \\tilde{u}\\frac{\\sin\\pi v_1}{\\pi}\n  \\quad {\\rm as} \\quad \\sigma \\to v_1 ,\n\\end{equation*}\n(\\ref{PIII_tExp_0}) reads\n\\begin{multline}\n  \\tau_{\\rm III'}(t) \\sim t^{(v_1^2-v^2_2)\/4}\n       \\Bigg\\{ 1 + \\frac{v_1v_2-v_1^2}{2v_1^2}t\n   \\\\\n   + \\tilde{u}\n      \\frac{\\sin\\pi(v_1-v_2)\/2}{\\sin\\pi v_1}\n      \\frac{\\Gamma(1-\\frac{\\displaystyle v_2-v_1}{\\displaystyle 2})\n            \\Gamma(1+\\frac{\\displaystyle v_2+v_1}{\\displaystyle 2})}\n           {\\Gamma^2(2+v_1)\\Gamma(1+v_1)}\n      \\left( \\frac{t}{4}\\right)^{1+v_1} \\Bigg\\} .\n\\label{PIII_tExp_0.2}\n\\end{multline}\nRecalling again (\\ref{HE_IIIparam}) and (\\ref{HE_tau}) we see that this agrees\nwith (\\ref{HE_Eexp}) provided\n\\begin{equation}\n  \\tilde{u}\\frac{\\sin\\pi\\mu}{\\sin\\pi(a+\\mu)}\n  = (1-\\xi)e^{\\pi i\\mu}-\\frac{\\sin\\pi a}{\\sin\\pi(a+\\mu)} ,\n\\label{HE_u}\n\\end{equation}\n(cf. (\\ref{LUE_u})).\n\n\\section{Application}\\label{S5}\n\\setcounter{equation}{0}\nIn a recent work relating to the application of random matrix theory to the\nstudy of moments of the derivative of the Riemann zeta-function, Conrey,\nRubinstein and Snaith \\cite{CRS_2005} obtained two asymptotic expressions\nassociated with the derivative of characteristic polynomials for random unitary\nmatrices. With $ U $ a Haar distributed element of the unitary group $ U(N) $,\nand $ e^{i\\theta_1},\\ldots,e^{i\\theta_N} $ its eigenvalues, let\n\\begin{equation}\n  \\Lambda_A(s) = \\prod^N_{j=1}(1-se^{-i\\theta_j}) ,\n\\label{UcharP}\n\\end{equation}\nand\n\\begin{equation}\n  {\\mathcal Z}_A(s) = e^{-\\pi iN\/2}e^{i\\sum^N_{n=1}\\theta_n\/2}s^{-N\/2}\\Lambda_A(s) ,\n\\label{UcharZ}\n\\end{equation}\n(note that $ {\\mathcal Z}_A(e^{i\\theta}) $ is real for $ \\theta $ real). In terms\nof this notation, the two results from \\cite{CRS_2005} are\n\\begin{equation}\n  \\langle |\\Lambda'_A(1)|^{2k} \\rangle_{A\\in U(N)} \\mathop{\\sim}\\limits_{N \\to \\infty}\n  b_k N^{k^2+2k} ,\n\\label{LcharM.1}\n\\end{equation}\nwhere\n\\begin{multline}\n b_k = (-1)^{k(k+1)\/2} \\sum^{k}_{h=0}{{k}\\choose{h}}(k+h)!\n \\\\\n \\times\n [x^{k+h}]\\left( e^{-x}x^{-k^2\/2}\n          \\det[I_{\\alpha+\\beta-1}(2\\sqrt{x})]_{\\alpha,\\beta=1,\\ldots,k}\\right) ,\n\\label{LcharM.1aux}\n\\end{multline}\nand\n\\begin{equation}\n  \\langle |{\\mathcal Z}'_A(1)|^{2k} \\rangle_{A\\in U(N)} \\mathop{\\sim}\\limits_{N \\to \\infty}\n  b'_k N^{k^2+2k} ,\n\\label{ZcharM.1}\n\\end{equation}\nwhere\n\\begin{equation}\n b'_k = (-1)^{k(k+1)\/2} (2k)!\n [x^{2k}]\\left( e^{-x\/2}x^{-k^2\/2}\n         \\det[I_{\\alpha+\\beta-1}(2\\sqrt{x})]_{\\alpha,\\beta=1,\\ldots,k}\\right) .\n\\label{ZcharM.1aux}\n\\end{equation}\nIn (\\ref{LcharM.1aux}) and (\\ref{ZcharM.1aux}) the notation $ [x^p]f(x) $ denotes \nthe coefficient of $ x^p $ in $ f(x) $.\n\nThe relevance of these formulae to the present study is that the determinant \ntherein can be identified in terms of $ \\tilde{E}^{\\rm hard}_{2} $. Thus, we have\nshown in a previous study \\cite{FW_2002a} that for $ a \\in \\mathbb Z_{\\geq 0} $\n\\begin{equation}\n \\tilde{E}^{\\rm hard}_{2}(s;a,\\mu;\\xi=1)\n  = A(a,\\mu)\\left(\\frac{2}{\\sqrt{s}}\\right)^{a\\mu}e^{-s\/4}\n                           \\det[I_{\\mu+\\alpha-\\beta}(\\sqrt{s})]_{\\alpha,\\beta=1,\\ldots,a} .\n\\label{HE_bessel}\n\\end{equation}\nwhere\n\\begin{equation}\n  A(a,\\mu) = a!\\prod^a_{j=1}\\frac{(j+\\mu-1)!}{j!} .\n\\label{.1}\n\\end{equation}\nInterchanging row $ \\beta $ by row $ a-\\beta+1 $ ($ \\beta=1,\\ldots,a $ in order) \nwe see from this that\n\\begin{equation}\n\\begin{split}\n  b_k  & = \\frac{(-1)^k}{A(k,k)}\\sum^k_{h=0} {{k}\\choose{h}}(k+h)!\n          [x^{k+h}]\\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1)\n  \\\\\n  b'_k & = \\frac{(-1)^k}{A(k,k)}(2k)!\n          [x^{2k}]\\left(  e^{x\/2}\\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) \\right)\n\\end{split}\n\\label{:a}\n\\end{equation}\nNote that the Painlev\\'e ${\\rm III^{\\prime}}\\;$ parameters appearing in this solution are \n$ \\mu=a=k\\in\\mathbb N $ and $ \\mu+a=2k\\in2\\mathbb N $ and thus we are dealing with the exceptional \ncase of indeterminacy referred to in Section 2. However as was noted there the \ngeneric formulae still hold with to the modifications discussed and in particular \nthe $\\sigma$-function has a small argument expansion of a purely analytic form. \n\nFrom (\\cite{FW_2003b}) it is known that the determinants in (\\ref{LcharM.1aux})\nand (\\ref{ZcharM.1aux}) can also be expressed as a particular generalised \nhypergeometric function. Such an observation implies, for instance, that\n\\begin{equation}\n x^{-k^2\/2}\\det[I_{\\alpha+\\beta-1}(2\\sqrt{x})]_{\\alpha,\\beta=1,\\ldots,k} \n   = \\prod^k_{j=1}\\frac{j!}{\\Gamma(j+k)}\n     {{}^{\\vphantom{(1)}}_0}F^{(1)}_1(;2k;x_1,\\ldots,x_k)|_{x_j=x} ,\n\\label{.b}\n\\end{equation} \nwhere $ {{}^{\\vphantom{(1)}}_0}F^{(1)}_1(;c;x_1,\\ldots,x_k) $ has a series\ndevelopment about $ x_1,\\ldots,x_k=0 $ with an explicitly given coefficient for an\narbitrary term.\nHowever this is not a practical or efficient way to compute the coefficients \nrequired in (\\ref{LcharM.1aux}) or (\\ref{ZcharM.1aux}) for moderate or large $ k $\nas it involves the hook lengths of Young diagrams associated with the partitions\nof $ k $. \n\nAccording to (\\ref{HE_E}), (\\ref{PIII_sigma}) and (\\ref{HE_Sexp})\n\\begin{equation}\n \\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) \n = \\exp\\left( -\\int^{4x}_0\\frac{ds}{s}\\;(\\sigma_{\\rm III'}(s)+k^2)\\right) ,\n\\label{Dzeta_tau}\n\\end{equation}\nwhere $ \\sigma_{\\rm III'}(s) $ satisfies the particular $ \\sigma $-Painlev\\'e \n${\\rm III^{\\prime}}\\;$ equation\n\\begin{equation}\n  (s\\sigma''_{{\\rm III}'})^2 \n  + \\sigma'_{{\\rm III}'}(4\\sigma'_{{\\rm III}'}-1)(\\sigma_{{\\rm III}'}-s\\sigma'_{{\\rm III}'})\n  - \\frac{k^2}{16} = 0 ,\n\\label{Dzeta_PIIIsigma}\n\\end{equation}\nsubject to the boundary condition\n\\begin{equation}\n  \\sigma_{\\rm III'}(s) \\mathop{\\sim}\\limits_{s \\to 0} -k^2+\\frac{s}{8}+{\\rm O}(s^2),\n  \\quad k\\in\\mathbb N .\n\\label{Dzeta_PIIIBC}\n\\end{equation}\nSubstituting \n\\begin{equation}\n  \\sigma_{\\rm III'}(s) = \\eta(s)+\\frac{s}{8} ,\n\\label{Dzeta_xfm}\n\\end{equation}\n(\\ref{Dzeta_PIIIsigma}) reads\n\\begin{equation}\n  (s\\eta'')^2 + 4((\\eta')^2-\\frac{1}{64})(\\eta-s\\eta') - \\frac{k^2}{4^2} = 0 .\n\\label{Dzeta_PIII.2}\n\\end{equation}\nWe see immediately that $ \\eta(s) $ can be expanded in an even function of $ s $ about\n$ s=0 $,\n\\begin{equation}\n  \\eta(s) = \\sum^{\\infty}_{n=0}c_ns^{2n}, \\qquad c_0=-k^2,\n  \\quad k\\in\\mathbb N .\n\\label{Dzeta_Exp}\n\\end{equation}\nMoreover the coefficients can be computed by a recurrence relation.\n\n\\begin{proposition}\\label{Dzeta_recur}\nSubstituting (\\ref{Dzeta_Exp}) in (\\ref{Dzeta_PIII.2}) shows\n\\begin{equation}\n  c_1 = \\frac{1}{64(4k^2-1)} ,\n\\label{Dzeta_c1}\n\\end{equation}\nwhile for $ p \\geq 2 $\n\\begin{multline}\n c_p = \\frac{1}{2c_1p(2p-1)+(2p-1)\/64-8pk^2c_1}\n \\\\\n \\times\\Bigg( 4k^2\\sum^{p-2}_{l=1}(l+1)(p-l)c_{l+1}c_{p-l}\n \\\\\n             -\\sum^{p-2}_{l=1}(l+1)(p-l)(2l+1)(2p-2l+1)c_{l+1}c_{p-l}\n \\\\\n             -\\sum^{p-1}_{l=1}(1-2l)c_{l}A_{p-l-1} \\Bigg) ,\n\\label{Dzeta_cp}\n\\end{multline}\nwhere\n\\begin{equation}\n  A_q = \\sum^{q}_{l=0}(l+1)(q-l+1)c_{l+1}c_{q-l+1} .\n\\label{Dzeta_aux}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWith $ h_l := (l+1)(2l+1)c_{l+1} $ we see\n\\begin{equation}\n  (s\\eta'')^2 = 4\\sum^{\\infty}_{p=1}H_{p-1}s^{2p}, \\qquad\n  H_p = \\sum^{p}_{l=0}h_lh_{p-l} ,\n\\label{Dzeta_aux1}\n\\end{equation}\nand similarly with $ a_l := (l+1)c_{l+1} $ we have\n\\begin{equation*}\n  (s\\eta')^2 = 4s^2\\sum^{\\infty}_{p=0}A_{p}s^{2p}, \\qquad\n  A_p = \\sum^{p}_{l=0}a_la_{p-l} .\n\\end{equation*}\nIt follows from this latter result that\n\\begin{equation}\n  \\left( (\\eta')^2-\\frac{1}{64} \\right)(\\eta-s\\eta')  = \\sum^{\\infty}_{p=0}G_{p}s^{2p} ,\n\\label{Dzeta_aux2}\n\\end{equation}\nwhere\n\\begin{equation*}\n  G_p = \\sum^{p}_{l=0}(1-2l)c_lb_{p-l}, \\quad\n  b_0 = -\\frac{1}{64}, \\quad b_p = 4A_{p-1} \\quad (p\\geq 1) .\n\\end{equation*}\nSubstituting (\\ref{Dzeta_aux1}) and (\\ref{Dzeta_aux2}) in (\\ref{Dzeta_PIII.2})\nand equating like coefficients of $ s^{2p} $ to zero shows that for $ p\\geq 1 $\n\\begin{equation*}\n  H_{p-1}+G_{p} = 0 .\n\\end{equation*}\nThis for $ p=1 $ implies (\\ref{Dzeta_c1}), and for $ p>1 $ implies (\\ref{Dzeta_cp}).\n\\end{proof}\n\nUsing Proposition \\ref{Dzeta_recur} it is straightforward to calculate, via \ncomputer algebra, the first $ k $ coefficients in (\\ref{Dzeta_Exp}) for any \nparticular value of $ k $. Furthermore use of computer algebra gives the \npower series up to $ x^{2k} $ of \n\\begin{equation*}\n  \\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) \\quad{\\rm and}\\quad\n  e^{x\/2}\\tilde{E}^{\\rm hard}_{2}(4x;k,k;\\xi=1) ,\n\\end{equation*}\naccording to (\\ref{Dzeta_tau}). From these power series the formulae \n(\\ref{:a}) are used to compute $ b_k $ and $ b'_k $. In \\cite{CRS_2005} the\nfirst 15 values of both $ b_k $ and $ b'_k $ were tabulated. This can be rapidly\nextended using the present method. However the resulting rational numbers \nquickly become unwieldy to record. Let us then be content by presenting just the\n16th member of the sequences,\n\\begin{equation*}\n  b_{16} = \\frac{\\scriptstyle \n307\n\\cdot\n23581\n\\cdot\n92867\n\\cdot\n760550281759\n             }\n             {\\scriptstyle \n2^{272}\n\\cdot\n3^{130}\n\\cdot\n5^{66}\n\\cdot\n7^{42}\n\\cdot\n11^{24}\n\\cdot\n13^{21}\n\\cdot\n17^{16}\n\\cdot\n19^{14}\n\\cdot\n23^{10}\n\\cdot\n29^{6}\n\\cdot\n31^{5}\n\\cdot\n37^{3}\n\\cdot\n41^{2}\n\\cdot\n43^{2}\n\\cdot\n47\n\\cdot\n53\n\\cdot\n59\n\\cdot\n61\n             } ,\n\\end{equation*}\n\\begin{equation*}\n  b'_{16} = \\frac{\\scriptstyle \n4148297603\n\\cdot\n7623077808870586151748455369217213506671334530597\n              }\n              {\\scriptstyle\n2^{264}\n\\cdot\n3^{133}\n\\cdot\n5^{66}\n\\cdot\n7^{42}\n\\cdot\n11^{25}\n\\cdot\n13^{21}\n\\cdot\n17^{16}\n\\cdot\n19^{14}\n\\cdot\n23^{11}\n\\cdot\n29^{7}\n\\cdot\n31^{6}\n\\cdot\n37^{3}\n\\cdot\n41^{2}\n\\cdot\n43^{2}\n\\cdot\n47\n\\cdot\n53\n\\cdot\n59\n\\cdot\n61\n              } .\n\\end{equation*}\n\n\\section{Acknowledgements}\nThis work was supported by the Australian Research Council. PJF thanks M. Rubinstein\nfor relating the results of \\cite{CRS_2005} before publication, and for the \norganisers of the Newton Institute program `Random matrix approaches in number theory'\nheld in the first half of 2004 for making this possible. NSW wishes to thank the \norganisers of the CRM program `Random Matrices, Random Processes and Integrable Systems'\nheld in Montreal 2005 for the opportunity to attend the program and in particular the \nhospitality of John Harnad during his stay.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nIn the field of machine learning, when there is a large number of features, determining the smallest subset that exhibits the strongest effect often decreases model complexity and increases prediction accuracy.\nThis process is called feature selection.\\footnote{Feature selection is also known as variable selection, attribute selection, and variable subset selection.} There are two well-known types of feature selection methods: filters and wrappers.\nFilters select features based on their relevance to the response variable independently of the prediction model.\nWrappers select features that increase the prediction accuracy of the model.\nHowever, as with any design decision during the construction of a prediction model, one needs to evaluate different feature selection methods in order to choose one, and above all to assess whether it is needed or not.\n\nWhile there has been extensive research on the impact of feature selection on prediction models in different domains, our investigation reveals that it is a rarely studied topic in the domain of bug prediction.\nFew studies explore how feature selection affects the accuracy of classifying software entities into buggy or clean~\\cite{Shiv13a}\\cite{Gao2011a}\\cite{Cata09b}\\cite{Kris11a}\\cite{Wang12a}\\cite{Khos10a}\\cite{Khos14a}\\cite{Ghot17a}, but to the best of our knowledge no dedicated study exists on the impact of feature selection on the accuracy of predicting the number of bugs.\nAs a result of this research gap, researchers often overlook feature selection and provide their prediction models with all the metrics they have on a software project or in a dataset.\nWe argue that feature selection is a mandatory step in the bug prediction pipeline and its application might alter previous findings in the literature, especially when it comes to comparing different machine learning models or different software metrics.\n\nIn this paper we treat bug prediction as a regression problem where a bug predictor predicts the number of bugs in software entities as opposed to classifying software entities as buggy or clean.\nWe investigate the impact of filter and wrapper feature selection methods on the prediction accuracy of five machine learning models: K-Nearest Neighbour, Linear Regression, Multilayer Perceptron, Random Forest, and Support Vector Machine.\nMore specifically, we carry out an empirical study on five open source Java projects: Eclipse JDT Core, Eclipse PDE UI, Equinox, Lucene, and Mylyn to answer the following research questions:\n\\vspace{0.2cm}\n\\\\\\emph{RQ1: How does feature selection impact the prediction accuracy?}\nOur results show that applying correlation-based feature selection (CFS) improves the prediction accuracy in 32\\% of the experiments, degrades it in 24\\%, and keeps it unchanged in the rest.\nOn the other hand, applying the wrapper feature selection method improves prediction accuracy by up to 33\\% in 76\\% of the experiments and never degrades it in any experiment. However, the impact of feature selection varies depending on the underlying machine learning model as different models vary in their sensitivity to noisy, redundant, and correlated features in the data.\nWe observe zero to negligible effect in the case of Random Forest models.\n\\vspace{0.2cm}\n\\\\\\emph{RQ2: Are wrapper feature selection methods better than filters?}\nWrapper feature selection methods are consistently either better than or similar to CFS.\nApplying wrapper feature selection eliminates noisy and redundant features and keeps only relevant features for that specific project, increasing the prediction accuracy of the machine learning model.\n\\vspace{0.2cm}\n\\\\\\emph{RQ3: Do different methods choose different feature subsets?}\nWe realize there is no optimal feature subset that works for every project and feature selection should be applied separately for each new project.\nWe find that not only different methods choose different feature subsets on the same projects, but also the same feature selection method chooses different feature subsets for different projects.\nInterestingly however, all selected feature subsets include a mix of change and source code metrics.\n\\vspace{0.2cm}\n\nIn summary, this paper makes the following contributions:\n\\begin{enumerate}\n\\item A detailed comparison between filter and wrapper feature selection methods in the context of bug prediction as a regression problem.\n\\item A detailed analysis on the impact of feature selection on five widely-used machine learning models in the literature.\n\\item A comparison between the selected features by different methods.\n\\end{enumerate}\n\nThe rest of this paper is organized as follows: In \\autoref{background}, we give a technical background about feature selection in machine learning.\nWe motivate our work in \\autoref{motivation}, and show how we are the first to study wrapper feature selection methods when predicting the number of bugs.\nIn \\autoref{empirical}, we explain the experimental setup, discuss the results of our empirical study, and elaborate on the threats to validity of the results.\nFinally, we discuss the related work in \\autoref{relatedWork} showing how our findings are similar or different from the state of the art, then conclude this paper in \\autoref{conclusions}.\n\n\n\n\\begin{figure}\n\\center{\\includegraphics[width=1.0\\linewidth]{.\/img\/biasvariance.pdf}}\n\\caption{The relationship between model complexity and model error \\cite{BiasVariance}.}\n\\label{fig:complexity}\n\\end{figure}\n\n\\section{Technical Background}\n\\label{background}\nTrained on bug data and software metrics, a bug predictor is a machine learning model that predicts defective software entities using software metrics.\nThe software metrics are called the independent variables or the features.\nThe prediction itself is called the response variable or the dependent variable.\nIf the response variable is the absence\/presence of bugs then bug prediction becomes a classification problem and the machine learning model is called a classifier.\nIf the response variable is the number of bugs in a software entity then bug prediction is a regression problem and the model is called a regressor.\n\nFeature selection is an essential part in any machine learning process.\nIt aims at removing irrelevant and correlated features to achieve better accuracy, build faster models with stable performance, and reduce the cost of collecting features for later models.\nModel error is known to be increased by both noise \\cite{Atla11a} and feature multicollinearity \\cite{Alle97c}.\nDifferent feature selection algorithms eliminate this problem in different ways.\nFor instance, correlation based filter selection chooses features with high correlation with the response variable and low correlation with each other.\n\nAlso when we build a prediction model, we often favour less complex models over more complex ones due to the known relationship between model complexity and model error, as shown in Figure \\autoref{fig:complexity}.\nFeature selection algorithms try to reduce model complexity down to the sweet spot where the total error is minimal.\nThis point is called the optimum model complexity.\nModel error is computed via the mean squared error (MSE) as: \n\\\\$MSE = \\frac{1}{N} \\sum_{i=1}^{N} (\\hat{Y}_i - Y_i)^2$ \\\\where $\\hat{Y}_i$ is the predicted value and $Y_i$ is the actual value.\nMSE can be decomposed into model bias and model variance as: \\\\$MSE = Bias^2 + Variance + Irreducible Error $ \\cite{Hast05a}\n\nBias is the difference between the average prediction of our model to the true unknown value we are trying to predict.\nVariance is the variability of a model prediction for a given data point.\nAs can be seen in Figure \\autoref{fig:complexity}, reducing model complexity increases the bias but decreases the variance.\nFeature selection sacrifices a little bit of bias in order to reduce variance and, consequently, the overall MSE.\n\nEvery feature selection method consists of two parts: a search strategy and a scoring function.\nThe search strategy guides the addition or removal of features to the subset at hand and the scoring function evaluates the performance of that subset.\nThis process is repeated until no further improvement is observed.\n\n\\begin{table}\n\\scriptsize\n\\caption{The CK Metrics Suite \\cite{Chid94a} and other object-oriented metrics included as the source code metrics in the bug prediction dataset \\cite{DAmb10c}}\n\\begin{center}\n\\begin{tabular}{ll} \n{Metric Name}\t\t& {Description}\\\\ \\hline\nCBO   & Coupling Between Objects \\\\[0.05cm] \nDIT   & Depth of Inheritance Tree \\\\[0.05cm] \nFanIn   & Number of classes that reference the class \\\\[0.05cm] \nFanOut   & Number of classes referenced by the class \\\\[0.05cm] \nLCOM   & Lack of Cohesion in Methods \\\\[0.05cm] \nNOC   & Number Of Children \\\\[0.05cm] \nNOA   & Number Of Attributes in the class \\\\[0.05cm] \nNOIA   & Number Of Inherited Attributes in the class \\\\[0.05cm] \nLOC   & Number of lines of code \\\\[0.05cm] \nNOM   & Number Of Methods \\\\[0.05cm] \nNOIM   & Number of Inherited Methods \\\\[0.05cm] \nNOPRA   & Number Of PRivate Atributes \\\\[0.05cm] \nNOPRM   & Number Of PRivate Methods \\\\[0.05cm] \nNOPA   & Number Of Public Atributes \\\\[0.05cm] \nNOPM   & Number Of Public Methods \\\\[0.05cm] \nRFC   & Response For Class \\\\[0.05cm] \nWMC   & Weighted Method Count  \\\\\\hline\n\n\\label{tbl:sourceMetrics}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\scriptsize\n\\caption{The change metrics proposed by Moser \\emph{et al.}\\xspace \\cite{Mose08a} included in the bug prediction dataset \\cite{DAmb10c}}\n\\label{tbl:changeMetrics}\n\\begin{center}\n\\begin{tabularx}{0.48\\textwidth}{lX} \n{Metric Name}\t\t& {Description}\\\\ \\hline\nREVISIONS   & Number of reversions \\\\[0.05cm] \nBUGFIXES   & Number of bug fixes \\\\[0.05cm] \nREFACTORINGS   & Number Of Refactorings \\\\[0.05cm] \nAUTHORS   & Number of distinct authors that checked a file into the repository \\\\[0.05cm] \nLOC\\_ADDED   & Sum over all revisions of the lines of code added to a file \\\\[0.05cm] \nMAX\\_LOC\\_ADDED   & Maximum number of lines of code added for all revisions \\\\[0.05cm] \nAVE\\_LOC\\_ADDED   & Average lines of code added per revision \\\\[0.05cm] \nLOC\\_DELETED   & Sum over all revisions of the lines ofcode deleted from a file \\\\[0.05cm] \nMAX\\_LOC\\_DELETED   & Maximum number of lines of code deleted for all revisions \\\\[0.05cm] \nAVE\\_LOC\\_DELETED   & Average lines of code deleted per revision \\\\[0.05cm] \nCODECHURN   & Sum of (added lines of code - deleted lines of code) over all revisions \\\\[0.05cm] \nMAX\\_CODECHURN   & Maximum CODECHURN for all revisions \\\\[0.05cm] \nAVE\\_CODECHURN   & Average CODECHURN for all revisions \\\\[0.05cm] \nAGE   & Age of a file in weeks (counting backwards from a specific release) \\\\[0.05cm] \nWEIGHTED\\_AGE   & Sum over age of a file in weeks times number of lines added during that week normalized by the total number of lines added to that file \\\\[0.05cm] \\hline\n\n\\end{tabularx}\n\\end{center}\n\\end{table}\n\n\n\\section{Motivation}\n\\label{motivation}\n\nIn this section, we shortly discuss the importance of predicting the number of bugs in software entities.\nThen, we highlight the impact of feature selection on bug prediction and particularly motivate the need for studying the wrapper methods.\n\n\\subsection{Regression vs Classification}\nMost of the previous research treats bug prediction as a classification problem where software entities are classified as either buggy or clean, and there have been several studies on the impact of feature selection on defect classification models.\nOn the other hand, bug prediction as a regression problem is not well-studied, and the effect of feature selection on predicting the number of bugs is not well-understood.\n\nSoftware bugs are not evenly distributed and tend to cluster \\cite{Ostr04a}, and some software entities commonly have larger numbers of bugs compared to others.\nPredicting the number of bugs in each entity provides valuable insights about the quality of these software entities~\\cite{Osma16c}, which helps in prioritizing software entities to increase the efficiency of related development tasks such as testing and code reviewing~\\cite{Khos03a}.\nThis is an important quality of a bug predictor especially for cost-aware bug prediction~\\cite{Mend09b}\\cite{Aris10a}\\cite{Kame10a}\\cite{Koba11a}\\cite{Hata12a}.\nIn fact, predicting the number of bugs in software entities and then ordering these entities based on bug density is the most cost-effective option \\cite{Osma17f}.\n\n\\subsection{Dimensionality Reduction}\n\\label{reduction}\n\nWhen the dimensionality of data increases, distances grow more and alike between the vectors and it becomes harder to detect patterns in data~\\cite{Bell60a}.\nFeature selection not only eliminates the confounding effects of noise and feature multicollinearity, but also reduces the dimensionality of the data to improve accuracy.\nHowever, feature selection does not seem to be considered as important as it should be in the field of bug prediction.\nFor instance, only 25 out of the 64 studied techniques in a recent research apply feature selection before training a machine learning model~\\cite{Malh15a}.\nOnly 2 out of the 25 are applied to bug prediction as a regression problem.\n\n\\subsection{Filters vs Wrappers}\n\\label{filtersWrappers}\nFeature selection methods are of two types: wrappers and filters \\cite{Koha97a}.\nWith wrappers, the scoring function is the accuracy of the prediction model itself.\nWrappers look for the feature subset that works best with a specific machine learning model.\nThey are called wrappers because the machine learning algorithm is wrapped into the selection procedure.\nWith filters (\\emph{e.g.},\\xspace CFS, InfoGain, PCA), the scoring function is independent of the machine learning model.\nThey are called filters because the attribute set is filtered before the training phase.\nGenerally, filters are faster than wrappers but less powerful because wrappers address the fact that different learning algorithms can achieve best performance with different feature subsets.\nIn this paper we aim at finding whether there is actually a difference between filters and wrappers in bug prediction, and then quantifying this difference.\n\nWrappers are known to be computationally expensive.\nThey become a bottleneck when the size of a dataset (features + data items) becomes large.\nHowever, this rarely happens in the bug prediction and bug prediction datasets tend to be relatively small.\nThis means that although wrappers are more resource intensive, they are easily applicable to bug prediction.\nNevertheless, our literature research yielded relatively few works that use wrappers for predicting number of bugs.\n\n\n\\section{Empirical Study}\n\\label{empirical}\n\nIn this section, we investigate the effect of feature selection on the accuracy of predicting the number of bugs in Java classes.\nSpecifically, we compare five widely-used machine learning models applied to five open source Java projects to answer the following research questions:\n\\\\\\emph{RQ1: How does feature selection impact the prediction accuracy?}\n\\\\\\emph{RQ2: Are wrapper feature selection methods better than filters?}\n\\\\\\emph{RQ3: Do different methods choose different feature subsets?}\n\n\\begin{table*}[h!]\n\\renewcommand{\\arraystretch}{1.0}\n\\footnotesize\n\\caption{Details about the systems in the studied dataset, as reported by D'Ambros \\emph{et al.}\\xspace~\\cite{DAmb10c}}\n\\begin{center}\n\\begin{tabular}{llrrrr} \n\t\t\t\t&\t\t&\t\t\t\t&\t\t\t&\t\t\t\t&\\% classes with more \\\\\nSystem \t\t\t&Release\t&KLOC\t\t\t&\\#Classes  \t& \\% Buggy\t\t&than one bug\t\\\\ \\hline\n\nEclipse JDT Core\t&3.4\t\t&$\\approx 224 $\t&997\t\t\t& $\\approx 20\\%$\t& $\\approx 7\\%$\\\\ \n\t\t\t\nEclipse PDE UI\t\t&3.4.1\t&$\\approx 40 $\t\t&1,497\t\t& $\\approx 14\\%$\t & $\\approx 5\\%$  \\\\ \n\t\t\t\nEquinox\t\t\t&3.4\t\t&$\\approx 39 $\t\t&324\t\t\t& $\\approx 40\\%$   \t& $\\approx 15\\%$ \\\\ \n\t\t\t\nMylyn\t\t\t&3.41\t&$\\approx 156$\t&1,862\t\t& $\\approx 13\\%$\t& $\\approx 4\\%$    \\\\ \n\t\t\t\nLucene\t\t\t&2.4.0\t&$\\approx 146$\t&691\t\t\t& $\\approx 9\\%$  \t& $\\approx 3\\%$\\\\ \\hline\n\\label{tbl:dataset}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\subsection{Experimental Setup}\n\\subsubsection*{Dataset}\nWe adopt the ``Bug Prediction Dataset\" provided by D'Ambros \\emph{et al.}\\xspace \\cite{DAmb10c} which serves as a benchmark for bug prediction studies.\nWe choose this dataset because it is the only dataset that contains both source code and change metrics at the class level, in total 32 metrics listed in~\\autoref{tbl:sourceMetrics} and \\autoref{tbl:changeMetrics}; and also provides the number of post-release bugs as the response variable for five large open source Java systems listed in \\autoref{tbl:dataset}.\nThe other dataset that has the number of bugs as a response variable comes from the PROMISE repository, but contains only 21 source code metrics \\cite{Jure10a}.\n \n\\subsubsection*{Prediction Models}\nWe use Multi-Layer Perceptron (MLP), Random Forest (RF), Support Vector Machine (SVM), Linear Regression (LR), and an implementation of the k-nearest neighbour algorithm called IBK.\nEach model represents a different category of statistical and machine learning models that is widely used in the bug prediction research~\\cite{Malh15a}.\n\nWe use the correlation-based feature selection (CFS) method \\cite{Hall00a}, the best \\cite{Chal08b}\\cite{Ghot17a} and the most commonly-used filter method in the literature \\cite{Malh15a}.\nFor the wrapper feature selection method we use the corresponding wrapper applicable to each prediction model.\nIn other words, we use MLP wrapper for MLP, RF wrapper for RF, SVM wrapper for SVM, LR wrapper for LR, and IBK wrapper for IBK.\nEvery feature selection method also needs a search algorithm.\nWe use the \\emph{Best First} search algorithm which searches the space of feature subsets using a greedy hill-climbing procedure with a backtracking facility.\nWe use this search algorithm because it returns the results in a reasonable amount of time while being exhaustive to a certain degree.\n \nWe use the Weka data mining tool \\cite{Hall09} to build prediction models for each project in the dataset.\nFollowing an empirical method similar to that of Hall and Holmes \\cite{Hall03a}, we apply each prediction model to three feature sets: the full set, the subset chosen by CFS, and the subset chosen by the wrapper.\nThe prediction model is built and evaluated following the 10-fold cross validation procedure.\nThe wrapper feature selection is applied using a 5-fold cross validation on the training set of each fold, then the best feature set is used.\nThe CFS algorithm is applied to the whole training set of each fold.\nThen the whole process is repeated 30 times.\nWe evaluate the predictions by means of the root mean squared error (RMSE).\nIn total, we have 25 experiments.\nEach experiment corresponds to a specific project and a specific prediction model trained on the three feature sets.\n\nWe use the default hyperparameter (\\emph{i.e.},\\xspace configuration) values of Weka 3.8.0 for the used machine learning models.\nAlthough hyperparameters can be tuned \\cite{Tant16a}\\cite{Osma17a}, we do not perform this optimization because we want to isolate the effect of feature selection.\nBesides, Linear Regression does not have hyperparameters and the gained improvement of optimizing SVM and RF is negligible \\cite{Tant16a}\\cite{Osma17a}.\n\n\n\n\\subsection{Results}\n\\autoref{fig:boxplots} shows standard box plots for the different RMSE values obtained by the different feature sets per prediction model per project.\nEach box shows 50\\% of that specific population.\\footnote{By population we mean the RMSE values of a specific experiment with a specific feature set.\nEach population consists of $10 \\times 50 = 500$ data items (10-fold cross validation done 50 times)} We can see that the wrapper populations are almost always lower than the full set ones, have smaller boxes, and have fewer outliers.\nThis means that applying the wrapper gives better and more consistent predictions.\nOn the other hand, we cannot make any observations about applying CFS because the difference between the CFS populations and the full set populations are not apparent.\n\n\\begin{figure*}\n\\center{\\includegraphics[width=1.0\\linewidth]{.\/img\/boxplots.pdf}}\n\\caption{Boxplots of all the experiments in our empirical study.\nThe y-axis represents the root mean squared error (RMSE).\nFor each project\/model, we examine three feature sets: the full set, the subset chosen by the CFS filter, and the subset chosen by the wrapper corresponding to the model.}\n\\label{fig:boxplots}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\center{\n\\includegraphics[width=1.0\\linewidth]{.\/img\/effect_size.pdf}}\n\\caption{This figure shows the bar plots of the effect size of the Dunn post-hoc analysis, which is carried out at the 95\\% confidence interval.\nThe x-axis indicates the pairwise comparison and the y-axis indicates the effect size.\nThe bars are color-coded.\nIf the bar is red, this means that the difference is not statistically significant.\nGrey means that there is a statistical significant difference, but the effect is negligible.\nBlue, golden, and green indicate a small, medium, and large statistically significant effect, respectively.}\n\\label{fig:effectSize}\n\\end{figure*}\n\n\nWhile box plots are usually good to get an overview of the different populations and how they compare to each other, they do not provide any statistical evidence.\nTo get more concrete insights, we follow the two-stage statistical test: Kruskal-Wallis + Dunn post-hoc analysis, both at the 95\\% confidence interval.\nWe apply the Kruskal-Wallis test on the results to determine whether different feature subsets have different prediction accuracies (\\emph{i.e.},\\xspace different RMSE).\nOnly when this test indicates that the populations are different, can we quantify such differences with a post-hoc analysis.\nWe perform Dunn post-hoc pairwise comparisons and analyze the effect size between each two populations.\n\\autoref{fig:effectSize} shows on the y-axis the detailed effect size between the two compared RMSE populations on the x-axis.\nIn this plot, there are two possible scenarios:\n\\begin{enumerate}\n\\item The Kruskal-Wallis test indicates that there is no statistical difference between the populations.\nThen all the bars are red to show that there is no effect between any two populations.\n\\item The Kruskal-Wallis test indicates a statistically significant difference between the populations.\nThen the color of the bars encode the pairwise effect size.\nRed means no difference and the two populations are equivalent.\nGrey means that there is a significant difference but can be ignored due to the negligible effect size.\nBlue, golden, and green mean small, medium, and large effect size respectively.\n\\end{enumerate}\n\nTo see how feature selection methods impact the prediction accuracy (\\emph{RQ1}), we compare the RMSE values obtained by applying CFS and wrappers with those obtained by the full feature set.\nWe observe that the RMSE value obtained by the CFS feature subset is statistically lower than the full set in 8 experiments (32\\%),\\footnote{negative non-red effect size in \\autoref{fig:effectSize}} statistically higher in other 6 experiments (24\\%),\\footnote{positive non-red effect size in \\autoref{fig:effectSize}} and statistically equivalent in 11 experiments (44\\%).\\footnote{red effect size in \\autoref{fig:effectSize}} Although CFS can decrease the RMSE by 24\\% on average (MLP with Mylyn), it can increase it by up to 24\\% (SVM with Lucene).\nWe also notice that applying CFS is not consistent within experiments using the same model.\nIt does not always improve, or always degrade, or always retain the performance of any model throughout the experiments.\nWe conclude that CFS is unreliable and gives unstable results.\nFurthermore, even when CFS reduces the RMSE, the effect size is at most small.\n\nOn the other hand, the RMSE value of the wrapper feature subset is statistically lower than that of the full set in 19 experiments (76\\%)  and statistically equivalent in the rest.\nApplying the wrapper feature selection method can decrease RMSE of a model by up to 33\\% (MLP with Eclipse JDT).\nWe also observe that the impact of the wrapper feature selection method on the accuracy is different from one model to another.\nIt has a non-negligible improvement on the prediction accuracy of IBK, LR, MLP, RF, and SVM in 80\\%, 60\\%, 100\\%, 20\\%, and 20\\% of the experiments, respectively.\nThis is due to the fact that different machine learning models are different in their robustness against noise and multicollinearity.\nMLP, IBK, and LR were improved significantly almost always in our experiments.\nOn the other hand, SVM and RF were not improved as often, because they are known to be resistant to noise, especially when the number of features is not too high.\nRF is an ensemble of decision trees created by using bootstrap samples of the training data and random feature selection in tree induction \\cite{Brei01a}.\nThis gives RF the ability to work well with high-dimensional data and sift the noise away.\nThe SVM algorithm is also designed to operate in a high-dimensional feature space and can automatically select relevant features \\cite{Hsu03a}.\nIn fact, this might be the reason behind the proven record of Random Forest and Support Vector Machine in bug prediction \\cite{Guo04a}\\cite{Elis08a}.\n\n\n\\begin{table}[]\n\\renewcommand{\\arraystretch}{1.2}\n\\normalsize\n\\caption{The level of agreement between different feature selection methods in each project}\n\\begin{center}\n\\begin{tabular}{lll} \nProject \t\t\t& $k$ \t&Agreement \\\\ \\hline\nEclipse JDT Core\t&0.18\t& Slight\\\\ \n\t\t\t\nEclipse PDE UI\t\t&0.17\t & Slight   \\\\ \n\t\t\t\nEquinox\t\t\t&0.40\t  & Fair  \\\\ \n\t\t\t\nMylyn\t\t\t&0.08\t & Slight   \\\\ \n\t\t\t\nLucene\t\t\t&0.18\t& Slight  \\\\ \\hline\n\\label{tbl:kappaOfProjects}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[]\n\\renewcommand{\\arraystretch}{1.2}\n\\normalsize\n\\caption{The level of agreement between the feature subsets selected by each method over all projects}\n\\begin{center}\n\\begin{tabular}{lll} \nFeature Selection Method \t\t\t& $k$ \t&Agreement \\\\ \\hline\nCFS\t\t\t\t\t\t\t\t&0.23\t& Fair\\\\ \n\t\t\t\nIBK Wrapper\t\t\t\t\t\t&0.26\t & Fair   \\\\ \n\t\t\t\nLR Wrapper\t\t\t\t\t\t&0.16\t  & Slight  \\\\ \n\t\t\t\nMLP Wrapper\t\t\t\t\t\t&0.04\t & Slight   \\\\ \n\nRF Wrapper\t\t\t\t\t\t&0.04\t & Slight   \\\\ \n\t\t\t\nSVM Wrapper\t\t\t\t\t\t&-0.01\t& Poor  \\\\ \\hline\n\n\\label{tbl:kappaOfMethods}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe wrapper method is statistically better than CFS in 18 experiments, statistically equivalent in 6 experiments, and worse in one experiment, but with a negligible effect size.\nThese results along with the fact that CFS sometimes increases the RMSE, clearly show that the wrapper selection method is a better choice than CFS (\\emph{RQ2}).\n\n\\begin{figure*}[]\n     \\begin{center}\n        \\subfigure[]{%\n            \\label{fig:selectedFeaturesGrid}\n            \\includegraphics[width=0.9\\textwidth]{.\/img\/selectedFeaturesGrid2.pdf}}\n\n        \\subfigure[]{%\n            \\label{fig:selectedFeaturesCounts}\n            \\includegraphics[width=0.9\\textwidth]{.\/img\/selectedFeaturesCounts2.pdf}}\n    \\end{center}\n\\end{figure*}\n\\begin{figure*}[]\n     \\begin{center}\n\n \\subfigure[]{%\n            \\label{fig:selectedFeaturesPerProject}\n            \\includegraphics[width=1\\textwidth]{.\/img\/selectedFeaturesPerProject.pdf}}\n\n    \\end{center}\n    \\caption{Subfigure (a) shows the features selected by each method using the whole data of each project.\nSubfigure (b) shows the number of times each feature is selected out of the 30 (1 CFS feature set + 5 wrapper sets per project).\nThe more times a feature is selected the more important it is for making accurate predictions.\nSubfigure (c) shows how different selection methods vary in the number of selected features.\nDetails about the features (metrics) are in \\autoref{tbl:sourceMetrics} and \\autoref{tbl:changeMetrics}}\n   \\label{fig:selectedFeatures}\n\\end{figure*}\n\n\n\\autoref{fig:selectedFeatures} shows the details about the features selected by each method using the whole data of each project in the dataset.\nTo answer the third research question (\\emph{RQ3}), we use the Fleiss' kappa statistical measure \\cite{Flei71a} to evaluate the level of agreement between the different feature selection methods for each project and the level of agreement of each feature selection method over the different projects. The Fleiss' kappa value, called $k$, is interpreted as follows:\n\\\\$k\\leq0 \\implies$ poor agreement\n\\\\$0.01 <k \\leq 0.20 \\implies$ Slight agreement\n\\\\$0.21 <k \\leq 0.40 \\implies$ Fair agreement\n\\\\$0.41 <k \\leq 0.60 \\implies$ Moderate agreement\n\\\\$0.61 <k \\leq 0.80 \\implies$ Substantial agreement\n\\\\$0.81 <k \\leq 1.00 \\implies$ Almost perfect agreement\n\n\nFigure \\autoref{fig:selectedFeaturesGrid} shows that different methods choose different features in each project.\nThe level of agreement between the different methods is \\emph{slight} in four projects and \\emph{fair} in only one, as detailed in \\autoref{tbl:kappaOfProjects}.\nAlso the same method chooses different features in different projects.\n\\autoref{tbl:kappaOfMethods} shows that the level of agreement between the feature subsets selected by the same method in different projects is at most \\emph{fair}.\nHowever, there exists some agreement on some features.\nFigure \\autoref{fig:selectedFeaturesCounts} shows that \\emph{REVISIONS, FanOut, NOIM,} and \\emph{FanIn} are chosen most often.\n\\emph{REVISIONS} in particular is chosen by all methods almost all the time.\nIt is selected in 28 out of 30 feature subsets,\\footnote{For each one of the 5 projects in the dataset, there are 6 feature subsets: 1 CFS subset and 5 wrapper subsets.} meaning that it has a high predictive power.\nOn the other hand, \\emph{RFC, MAX\\_LOC\\_DELETED, NOM}, and \\emph{LCOM} are picked the least number of times, which means they have little to no predictive power.\n\nThe number of discarded features varies from 10 to 28 features out of 32 in total, as detailed in Figure \\autoref{fig:selectedFeaturesPerProject}.\nThis means that most of the features can be removed while enhancing (in 48\\% of the experiments), or at least retaining (in 52\\% of the experiments), the prediction accuracy.\n\nAnother important observation is that no feature subset contains only source code metrics or only change metrics, but always a mix of both.\nThis means that no category of metrics (\\emph{i.e.},\\xspace change and source code) alone is good for predicting the number of bugs, but they should be combined to achieve better performance.\nPrevious studies show that change metrics are better than source code metrics in bug prediction and combining the two sets either does not bring any benefit \\cite{Mose08a} or hinders the performance \\cite{Aris10a}.\nHowever, these studies either did not employ feature selection at all \\cite{Mose08a} or employed only CFS \\cite{Aris10a}.\n\n\n\\subsection{Threats to Validity}\n\nAlthough we use a well-known benchmark as our dataset, the quality of our results is very dependent on the quality of that dataset.\nAlso our dependence on WEKA for building the machine learning models, makes the quality of the models dependent solely on the quality of the WEKA implementation itself.\nSince we are dealing with machine learning, the risk of overfitting always exists.\nIn general, we mitigate this risk by repeating each feature selection\/training\/testing process 50 times.\nAlso for the wrapper method, we apply the train-test-validate (TTV) approach to mitigate any overfitting bias in favour of wrappers.\\footnote{The train-test-validate is not applicable to CFS due to how CFS operates} \n\nIn our study, the dataset contains metrics only from open source software systems.\nIt is hard to generalize the results to all systems due to the differences between industrial and open source projects.\nAlso all the systems in the dataset are developed in Java and the findings might not generalize to systems developed in other languages.\nAnother important threat to external validity comes from the chosen prediction models.\nAlthough we try to pick a representative set of prediction models, our findings might not generalize to other machine learning models such as hybrid and genetic ones.\n\nMachine learning methods depend on the underlying data.\nAlthough our findings are statistically strong, they may not generalize to all open source Java systems.\nPractitioners and researchers are encouraged to run comparative studies, similar to ours, before making decisions regarding the machine learning techniques employed for bug prediction.\n\n\\section{Related Work}\n\\label{relatedWork}\n\nIn a recent systematic literature review of the machine learning techniques in the bug prediction field~\\cite{Malh15a}, Most studies (60\\%) do not apply feature selection at all.\nOur investigation shows that researchers in bug prediction often undermine the importance of feature selection.\n\nAmong the studies that apply feature selection, CFS \\cite{Hall00a} is the most commonly used \\cite{Malh15a}.\nMany studies employ the CFS filter technique before training machine learning models \\cite{Elis08a}\\cite{Kaur08a}\\cite{Aris10a} \\cite{DeCa10a}\\cite{Malh11a}\\cite{Zimm09a}\\cite{Twal11a}\\cite{Okut14a}.\nWe also apply CFS as the baseline filter technique and compare it to wrapper techniques to show that wrappers outperform this filter in most cases.\n\nMost studies that apply feature selection actually apply filter techniques like principal component analysis (PCA) \\cite{Khos06a}\\cite{DAmb10c}\\cite{Naga06a}\\cite{Kanm07a}\\cite{Sing10a}, consistency based selection (CBS) \\cite{Zimm09a}, and InfoGain \\cite{Menz07a}\\cite{Turh07a}\\cite{Sing09a}\\cite{Misi11a}.\nVery few studies apply wrapper feature selection techniques \\cite{Cahi13a}\\cite{Gond08a} when classifying software entities as buggy or clean (classification).\nHowever, to the best of our knowledge, there is no study that applies wrapper techniques when predicting the number of bugs in software entities (regression).\nIn our study we compare the CFS filter with different wrappers while treating bug prediction as a regression problem.\n\nShivaji \\emph{et al.}\\xspace \\cite{Shiv13a} study the impact of multiple feature selection techniques (filters and wrappers) on the performance of Na\\\"ive Bayes and Support Vector Machines when classifying code changes as buggy or clean.\nThey report a significant enhancement in the accuracy measures of both classifiers when feature selection is applied.\nHowever, Shivaji \\emph{et al.}\\xspace consider \\emph{``everything in the source code separated by whitespace or a semicolon\"} as a feature \\cite{Shiv13a}.\nThis includes variable names, method names, keywords, comment words, \\emph{etc.}\\xspace\nThey end up with a staggering number of features ranging from 6,127 to 41,942 features.\nThe vast majority of features in the initial feature set are bound to be irrelevant, hence the results of their study are exaggerated and cannot be generalized.\n\nChallagulla \\emph{et al.}\\xspace \\cite{Chal08b} report that performing principal component analysis before training the models does not result in a significant improvement of the performance of the bug predictors, while correlation-based feature selection (CFS) and consistency-based subset evaluation (CBS) actually decrease the prediction error of the machine learning models.\nWe actually report different results when applying CFS to regressors.\n\nKhoshgoftaar \\emph{et al.}\\xspace \\cite{Khos10a} combine filter feature selection and data sampling to achieve better classification accuracy on the package level.\nThey report an improvement of around 2\\% to the area under the receiver operator characteristic curve (AUC-ROC) when the filters are applied to sampled data rather than the whole data.\nThen Khoshgoftaar \\emph{et al.}\\xspace \\cite{Khos14a} improve the performance by repeatedly applying the sampling and filtering several times then aggregating the results.\n\nIn their defect-proneness prediction framework, Song \\emph{et al.}\\xspace \\cite{Song11a} included a special step for feature selection.\nHowever, their framework treats bug prediction as a classification problem.\nIn this paper we study the effect of wrapper feature selection methods on bug prediction as a regression problem.\n\nGao \\emph{et al.}\\xspace \\cite{Gao2011a} studied seven filter feature selection techniques.\nThey report that the studied classification models were either improved or remained unchanged while 85\\% of the original features were eliminated.\nKrishnan \\emph{et al.}\\xspace \\cite{Kris11a} analyze whether change metrics remain good predictors during the evolution of Eclipse.\nIn their analysis, they use J48 decision tree as the machine learning algorithm and refer to the top five features as the set of prominent predictors, then study the consistency of this set over the consecutive versions of Eclipse.\nThey report that there is a small subset of change metrics that is consistently good at classifying software entities as buggy or clean across products and revisions.\nWang \\emph{et al.}\\xspace \\cite{Wang12a} study the effect of removing redundant or useless features from the PROMISE dataset.\\footnote{\\url{http:\/\/openscience.us\/repo\/}}\nThey report that feature selection improves classification accuracy.\nCatal and Diri \\cite{Cata09b} explore which machine learning algorithm performs best before and after applying feature reduction.\nIn these studies, researchers treat bug prediction as a classification problem while we study the effect of feature selection on bug prediction as a regression problem.\n\nTurhan and Bener \\cite{Turh09a} argue that Na\\\"ive Bayes assumes the independence and the equal importance of features.\nThese assumptions are not true in the context of bug prediction.\nYet, Na\\\"ive Bayes is one of the best classifiers \\cite{Menz07a}.\nThey empirically show that the independence assumption is not harmful for defect prediction using Na\\\"ive Bayes and assigning weights to features increases the performance and removes the need for feature selection.\nThey conclude that \\emph{``either weighted Naive Bayes or pre-processing data with PCA may produce better results to locate software bugs in source code\"}\\cite{Turh09a}.\n\nWe confirm that most of the existing literature treats bug prediction as a classification problem, and that studying the number of bugs is neglected in the field.\nFor instance, Ostrand \\emph{et al.}\\xspace \\cite{Ostr05a} use negative binomial regression (NBR) to predict the number of bugs in software modules.\nThey report that NBR fits the bug prediction problem and demonstrates high accuracy.\nJanes \\emph{et al.}\\xspace \\cite{Jane06a} compare three count models to predict the number of bugs and find out that zero-inflated NBR performs better than Poisson regression and NBR.\nRathore and Kumar \\cite{Rath16a} investigate six different fault prediction models for predicting the number of bugs and show that count models (\\emph{i.e.},\\xspace NBR and Zero-Inflated Poisson regression) underperform compared to linear regression, decision tree regression, genetic programming and multilayer perceptron.\nGraves \\emph{et al.}\\xspace \\cite{Grav00a} build a generalized linear regression model to predict the number of bugs based on the various change metrics.\nGao and Khoshgoftaar \\cite{Gao07a} compare the performance of several count models (\\emph{e.g.},\\xspace Poisson regression) and show that Hurdle Poisson regression has the most accurate predictions.\nNevertheless, these studies do not apply feature selection and their results should be reassessed in the light of our findings.\n\n\n\\section{Conclusions}\n\\label{conclusions}\nGeneralizing bug prediction findings is hard.\nSoftware projects have different teams, cultures, frameworks, and architectures.\nConsequently, software metrics have different correlations with the number of bugs in different projects.\nThese correlations can be captured differently by distinct prediction models.\nWe argue that wrapper feature selection methods fit this problem best because they not only choose features relevant to the response variable but also to the prediction model itself.\nIndeed, our results show that wrapper feature selection is always better than CFS and improves the performance of a model (by up to 47\\%) while eliminating most of the features (up to 87\\%).\nOur results also reveal that there is little agreement on the selected features.\nEven the same method chooses different features in different projects.\nWe cannot generalize what feature subset to use, but we can recommend combining both change and source code metrics and letting the wrapper feature selection method choose the right subset.\n\n\nIn the future, we plan to investigate how the selected feature subset changes with the evolution of a certain project from the size and selection perspectives.\nLastly, while carrying out this research we realized that datasets providing the number of bugs as the response variable are scarce, which could be a hurdle to studies predicting the number of bugs in software systems.\nWe therefore encourage the community to publish such defect datasets more frequently.\n\\section*{Acknowledgments}\nThe authors gratefully acknowledge the funding of the Swiss National Science Foundations for the project ``Agile Software Analysis\" (SNF project No. 200020\\_162352, Jan 1, 2016 - Dec. 30, 2018).\\footnote{\\url{http:\/\/p3.snf.ch\/Project-162352}}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn perturbative quantum field theory, the traditional method to compute\ncross sections and distributions for a physical process involves generating\ntree and loop amplitudes via Feynman diagrams and interfering them.\nSimplifications of the expressions are performed at the analytical level.\nHere an essential part is the reduction of the typically dimensionally regularized\nloop integrals~\\cite{'tHooft:1972fi} to a small number of standard integrals.\nThis step can be performed at the amplitude level for tensor integrals or,\nafter contraction of Lorentz indices, at the level of interferences for scalar\nintegrals. Considering the case of scalar integrals,  \nintegration by parts (IBP) identities~\\cite{Tkachov:1981wb,Chetyrkin:1981qh}\nand Lorentz invariance (LI) identities~\\cite{Gehrmann:1999as} may be used\nfor a systematic reduction to a set of independent integrals, called master\nintegrals.\nThe standard reduction algorithm by Laporta~\\cite{laporta} defines an\nordering for Feynman integrals, generates identities and solves the\nresulting system of linear equations.\nAlternative methods to exploit IBP and LI identities for reductions have been\nproposed~\\cite{Smirnov:2005ky,Smirnov:2006tz,Gluza:2010ws,Schabinger:2011dz},\nsee also \\cite{Lee:2008tj,Grozin:2011mt} and references therein.\nPublic implementations of different reduction algorithms are available with the\ncomputer programs AIR~\\cite{Anastasiou:2004vj}, FIRE~\\cite{Smirnov:2008iw}\nand the first version of \\reduze~\\cite{Studerus:2009ye}.\n\nThis article presents the new public reduction program \\reduze\\;2.\nIt is written in {\\tt C++} and represents a major rewrite and extension of its\npredecessor \\reduze. In the following, the name \\reduze\\ refers to the new\nversion presented here.\n\nIn \\reduze, integrals are indexed by integral families (``auxiliary topologies'')\nand sectors (``topologies'') therein.\nFor the reduction, the program implements a fully distributed variant of\nLaporta's algorithm using the Message Passing Interface (MPI).\nIn this way, not only different sectors can be reduced in parallel, but also\nthe integrals of a single sector can be reduced in a distributed computation.\n\nThe program allows to utilize multiple integral families within a calculation.\nSpecial emphasis has been placed on finding relations between sectors\nof the same or different integral families and employing them\nto eliminate integrals.\nBesides a straightforward combinatorial matcher, the program implements\ngraph and matroid theory based algorithms to compute such relations,\ntaking into account possible crossings of external momenta.\nSimilar to the program \\diana~\\cite{Tentyukov:1999yq}, \\reduze\\ may be used\nto shift loop momenta of Feynman diagrams generated by a program like\n\\qgraf~\\cite{qgraf} or \\feynarts~\\cite{Hahn:2000kx} to match sectors of\nintegral families.\n\nOther features include the generation of differential equations for Feynman\nintegrals and the computation of (bare) amplitude interferences up to master\nintegrals, starting from Feynman diagrams generated by \\qgraf.\nFor storing intermediate results of a reduction, optionally,\nthe transitional open source database \\berkeleydb~\\cite{berkeley}\ncan be used.\n\nFor the normalization of algebraic coefficients\nin the identities, one can choose between \\ginac~\\cite{Bauer:2000cp}\nand \\fermat~\\cite{fermat}.\nReduction identities and other results can be exported to\n\\form~\\cite{form}, \\mathematica~\\cite{mathematica}, and \\maple~\\cite{maple}\nformat.\nConfiguration and job files use the \\yaml\\ format~\\cite{yaml} and are parsed\nwith the {\\tt yaml-cpp} parser~\\cite{yamlcpp}.\n\n\\reduze\\;2 was used to calculate the two-loop leading color corrections\nto heavy-quark pair production in the gluon fusion channel~\\cite{Bonciani:2010mn}.\nLast but not least, \\reduze\\ is published as open source under the\n{\\tt GNU} General Public License (GPL) v3 and has no mandatory dependencies\non proprietary software.\n\n\n\\section{Basic concepts and notations}\n\n\\subsection{Integral families, sectors, and integrals}\n\\label{sec:fam_sec_int}\n\nA \\emph{propagator} $P$ is defined as the expression $1\/(q^2 - a)$\nwhere $q$ is a four-momentum and $a$ is constant.\nThe momentum $q$ of a propagator (defined up to a minus sign) is a\nlinear combination of loop momenta $k_i$ and external momenta $p_i$,\nand $q^2$ is a scalar product in Minkowski space with the metric convention\n$g = \\mbox{diag}(1, -1, -1, -1)$.\nIn \\reduze, also generalized propagators $1\/(q l - m^2)$\nwith the scalar product of two different momenta $q$ and $l$ are available\nto support more general irreducible numerators.\n\nAn $l$-loop \\emph{integral family} (or ``auxiliary topology'') $F$ is an\nordered set $\\{ P_1, \\ldots, P_n \\}$\nof propagators $P_i$, $i=1,\\ldots,n$, which is minimal and complete in the sense,\nthat any scalar product of a loop momentum $k_i$ with a loop momentum $k_j$\nor an external momentum $p_j$ can be uniquely expressed as a linear combination\nof inverse propagators and kinematic invariants.\nDenoting the number of independent\nexternal momenta by $m$, an integral family must contain exactly\n$l\\,(l+1)\/2+l\\,m$ propagators, where the first term counts the scalar products\nbetween loop momenta only and the second term the products involving both loop and\nexternal momenta.\nA new feature of \\reduze\\  is its ability to handle several integral families\nsimultaneously.\n\nA selection of $t$ propagators of an integral family defines a \\emph{sector} of\nthis family.\nAssuming a sector has the propagators $P_{j_1},\\ldots, P_{j_t}$ with\n$\\{j_1,\\ldots,j_t\\} \\subset \\{1,\\ldots,n\\}$, then its identification number\nis defined as\n\\begin{equation}\n ID = \\sum_{k=1}^{t} 2^{j_k-1} \\, .\n\\end{equation}\nThere are in general $\\binom{n}{t}$ different $t$-propagator sectors and\n$\\sum_{t=0}^{n} \\binom{n}{t} = 2^n$ sectors are contained in an integral\nfamily. Their identification numbers fulfill $0 \\leq ID \\leq 2^n - 1$ .\n\nA sector whose propagators form a subset of the propagators of another sector\nof the same integral family is a \\emph{subsector} of the other sector.\n\nThe purpose of an integral family is to index scalar loop integrals.\nTo every $t$-propagator sector with propagators $P_{j_1}$, $\\ldots$, $P_{j_t}$\nbelongs a infinite set of $d$-dimensionally regularized $l$-loop\nintegrals~\\cite{'tHooft:1972fi} which all share the same propagators.\nThese integrals have the generic form\n\\begin{equation}\\label{eq:dim_reg_int}\nI = \\int \\mbox{d}^d k_1 \\ldots \\int \\mbox{d}^d k_l \\,\nP_{j_1}^{r_1} \\ldots P_{j_t}^{r_t}\n     P_{j_{t+1}}^{-s_1} \\ldots P_{j_n}^{-s_{n-t}}\n\\end{equation}\nwith integer exponents $r_i \\geq 1$ and $s_i \\geq 0$.\nIn \\reduze\\  such an integral is represented by\n\\begin{equation}\\label{eq:int_reduze_format}\n\\mbox{I}(F, t, ID, r, s,\\{v_1,\\ldots,v_{n}\\})\n\\end{equation}\nwhere $F$ denotes the integral family,\n$r=\\sum_{i=1}^t r_i \\geq t$, $s=\\sum_{i=1}^{n-t} s_i \\geq 0$ and\n$v_i$ is the exponent of propagator $P_i$.\nPositive $v_i$ denote powers of regular propagators (non-trivial denominator),\nnegative $v_i$ denote powers of inverse propagators (non-trivial numerator),\nand zero means absence of a propagator.\nThe numbers $t$, $r$, $s$ as well as the identification number $ID$ of the\nsector, to which the integral belongs, can be calculated from the vector $v$.\n\nConsider a $t$-propagator sector of a $n$-propagator integral family.\nThe number of integrals that one can build for certain values of $r$ and $s$ is given by\n$\n{\\mathcal N}(n,t,r,s) = \\binom{r-1}{t-1} \\binom{s+n-t-1}{n-t-1} \\, .\n$\nThe two binomial factors count all possible ways to arrange the exponents of\nthe propagators in the denominator and numerator, respectively.\n\nThe integral with $r=t$ and $s=0$ of some sector is called \n\\emph{corner integral} of this sector.\n\n\n\n\\subsection{Integration by parts (IBP) identities}\nIn dimensional regularization~\\cite{'tHooft:1972fi} the integral over a total derivative\nis zero. Let ${\\mathbf I'}$ be the integrand of an integral of the form (\\ref{eq:dim_reg_int}).\nThen, working out the differentiation in\n\\begin{equation}\n \\int \\mbox{d}^d k_{i} \\, \\frac{\\partial}{\\partial k_{i}^{\\mu}} \\, \\big[ q^{\\mu} \\,\n {\\mathbf I'}(p_1, \\ldots, p_{m}, k_1, \\ldots, k_l) \\big] = 0\n\\end{equation}\n\nleads to the integration by parts (IBP) identities~\\cite{Tkachov:1981wb,Chetyrkin:1981qh}.\nThe momentum $q$ is an arbitrary loop or external momentum. The index $\\mu$ is summed over\nbut the index $i$ is not. If there are $l$ loop momenta and $m$ independent external\nmomenta one can therefore build $l\\, (l+m)$ equations from one integral, the\n\\emph{seed integral}.\n\n\n\\subsection{Lorentz invariance (LI) identities}\nOne can also use the Lorentz invariance of the integrals~\\cite{Gehrmann:1999as}. Taking\nan integral ${\\mathbf I}(p_{1},\\ldots ,p_{m})$ the following equation holds\n\\begin{equation}\n\\sum_{n=1}^m \\Big( p_{n}^{\\nu} \\frac{\\partial}{\\partial p_{n \\mu}}\n    - p_{n}^{\\mu} \\frac{\\partial}{\\partial p_{n \\nu}} \\Big)\\, {\\mathbf I}\n    (p_{1},\\ldots ,p_{m}) = 0 \\, .\n\\end{equation}\nThe derivatives can be shifted directly to the integrand of the integral ${\\mathbf I}$.\nThis equation can be contracted with all possible antisymmetric combinations of the\nexternal momenta, e.g.\\ $p_{1 \\mu} p_{2 \\nu} - p_{1 \\nu} p_{2 \\mu}$, which leads to\n$m\\,(m-1)\/2$ equations where $m$ denotes the number of independent external momenta.\nAs it was shown in~\\cite{Lee:2008tj} the LIs do not give new linearly independent\nequations in addition to the IBPs. However, they can accelerate the convergence in a\nreduction, since in general an LI identity generated from one seed integral cannot be\nreproduced with the IBP identities generated from the same seed integral alone.\n\\reduze\\  offers the possibility to use the LIs.\n\n\n\\subsection{Zero sectors}\n\nIt is possible that a whole sector is zero which means that all integrals belonging\nto this sector are zero. A sector of an $l$-loop integral family is trivially\nzero if it does not allow for a selection of $l$ propagator momenta which are\nindependent with respect to the $l$ loop momenta.\nThe graph based methods in \\reduze, see section \\ref{sec:physical_secs},\nwill automatically detect these cases.\nAs a second method, a sector is set to zero if the reduced IBP identities\ngenerated from the seed integrals of this sector with $r=t$ and\n$s=0, 1$ show that its corner integral is zero.\n\n\\subsection{Sector relations} \\label{sec:sec_rel}\n\nGiven a scalar loop integral as well as one or several integral families,\nsuppose it is possible to map the integral to a linear combination of indexed\nintegrals of type \\eqref{eq:int_reduze_format}.\nIn general, such a map is not unique.\nAmbiguities may arise if sectors from different integral families have the same\nset of propagators or if a transformation of loop and external momenta in\n\\eqref{eq:dim_reg_int} leads to a different linear combination of type\n\\eqref{eq:int_reduze_format}.\nFor the corner integral of some sector $S$ written in the form\n\\eqref{eq:dim_reg_int},\nconsider the transformation of integration variables\n\\begin{align}\\label{eq:shift_trafo}\nk_i &\\to \\sum_{j=1}^{l} M_{ij} k_j + \\sum_{j=1}^{m} N_{ij} p_j\n\\end{align}\nwith $|\\det{M}|=1$.\nIf the new integrand factors can be identified with propagators of a\nsector $S'$, the ``shift'' transformation \\eqref{eq:shift_trafo} defines a\n\\emph{sector relation} between $S$ and $S'$.\nIn this case, any integral in the sector $S$ can be expressed as a linear\ncombination of integrals in the sector $S'$ and subsectors of $S'$.\nIf $S$ and $S'$ are different, such a relation can be used to eliminate\none of the two sectors completely.\nThe case where $S$ and $S'$ are identical is discussed in\nsection~\\ref{sec:sec_symm}.\n\n\\reduze\\  is able to automatically detect sector relations or handle relations\nsupplied by the user.\nSector relations will be used to eliminate redundant sectors or integrals,\nusually at the earliest possible stage.\nAs a particularly useful special case, a shift \\eqref{eq:shift_trafo} might\nmap each propagator of an integral family to another propagator of the same\nintegral family.\nThis leads to a one--to--one mapping between integrals as well as sectors\nof the integral family.\nSuch a relation can be entered for the full integral family via permutations\nof propagators and allows for particularly efficient removal of redundancies.\n\nAs a generalization of the above described concepts,\n\\reduze\\  also allows for crossings of\nexternal momenta in addition to \\eqref{eq:shift_trafo}.\nIf the involved crossing leaves the kinematic invariants\nunchanged, the corresponding relations can be directly exploited for\nrelations between sectors as described before.\nIn the general case, \\reduze\\  also handles relations between sectors of\nintegral families defined with crossed kinematics.\n\n\\subsection{Sector symmetries}\n\\label{sec:sec_symm}\n\nSpecial shifts of the loop momenta as in \\eqref{eq:shift_trafo} which\ntransform a sector to itself are called \\emph{sector symmetries}.\nThese shifts are also allowed to contain a permutation of the external\nmomenta as long as it does not change the kinematic invariants.\nSector symmetries may be used to express integrals in terms of other\nintegrals in the same sector and its subsectors.\nThese relations may provide information complementary to the IBP and LI\nidentities and can be used in the reduction to find a minimal number of\nmaster integrals.\n\\reduze\\ is capable of automatically determining sector symmetries\nor handling user supplied rules, and offers to exploit them for reductions.\n\n\n\\section{Graph and matroid based algorithms for sectors}\n\n\\subsection{Physical sectors}\n\\label{sec:physical_secs}\n\nA physical sector is a sector whose propagators correspond to edges in\na graph such that momentum is conserved. The construction of a graph from\na sector, i.e. from the momenta of a set of propagators with $l$-loop momenta,\ncan be done by choosing $l$ propagators which have independent momenta with\nrespect to the loop momenta and identifying them as edges in a graph with\nboth ends glued together in a single root vertex. External edges are also\nattached to this root vertex with one of their ends.\nSubsequently, for each of the remaining propagators, a vertex of the graph\n(first the root vertex) is cleaved into two vertices, and a new edge is inserted\nbetween these vertices, such that the edge's momentum (determined by momentum\nconservation) exactly matches the propagator's momentum.\n\nWith this procedure \\reduze\\ automatically constructs graphs for sectors\nwhere this is possible and thus identifies physical sectors.\nThe possibility of having graph representations for sectors gives\naccess to fast algorithms for identifying isomorphic graphs and\nfinding sector relations and sector symmetries.\n\n\n\\subsection{Sector relations}\n\nIf graphs constructed for two different physical sectors are isomorphic,\na shift of the form \\eqref{eq:shift_trafo} between the two sets of loop momenta\ncan be derived by identifying the edges of the two graphs together\nwith their oriented momentum flow labeling.\n\\reduze\\ offers the possibility to find relations between physical sectors\nby this strategy, allowing also crossings of external legs for the graph\nisomorphism but restricting to cases with $|\\det{M}|=1$.\nFor the graph of each physical sector a standard form of its adjacencies,\na \\emph{canonical label}, is computed with the algorithm\n\\cite{McKay1981}.\nWe distinguish different masses by replacing massive edges with a chain\nof several edges, where the length of such a chain labels a mass uniquely.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{matroid1}\\hspace{20mm}\n\\includegraphics[width=0.25\\textwidth]{matroid2}\n\\end{center}\n\\caption{\\label{fig:twists}\nTwo non-isomorphic graphs which are related by a twist.\nThe matroids of these graphs are isomorphic.\n}\n\\end{figure}\n\nGraph isomorphism takes into account the ambiguities in labeling the\n\\emph{nodes} of a graph.\nWhile isomorphic graphs can be described by the same propagators (possibly\nwith a crossing of external momenta), the inverse is not true.\nConsider for instance the two vacuum graphs in figure~\\ref{fig:twists}.\nThe two graphs are non-isomorphic but can be described by the same\npropagators, i.e.\\ by the same sector.\nHere, a more appropriate object to consider is not the graph of the sector,\nbut the associated matroid.\nMatroids are based on the notion of a set of linearly independent sets\nand may be considered as generalizations of graphs.\nFor a graph an associated graph matroid (or cycle matroid) can be defined via\nits edges. Definitions and fundamental properties are given in the review\n\\cite{Bogner:2010kv} and references therein.\nEssential for us is the following chain of statements.\nThe relevant properties of a vacuum graph where all edges share a\ncommon mass is encoded in the first Symanzik polynomial\n($\\mathcal{U}$ polynomial).\nFor brevity of the argument let us furthermore restrict to biconnected graphs.\nThe generalization to arbitrary vacuum components with different masses is\nrather straightforward.\nAn immediate combinatorial approach to isomorphisms of the\nSymanzik polynomials, which is not restricted to vacuum graphs, was presented in\n\\cite{Pak:2011xt}.\nHere, we note that the first Symanzik polynomials\nof two graphs are equal up to a permutation of edge variables exactly if\ntheir matroids are isomorphic, see \\cite{Bogner:2010kv} and references therein.\nTwo matroids of biconnected vacuum graphs are isomorphic exactly if the\ntwo graphs are isomorphic up to a series of \\emph{twists}, a statement known\nas Whitney's theorem \\cite{whitney}.\nA twist operation starts by breaking a graph into two graphs such that\nidentification of \\emph{separation pairs} of nodes in both graphs restore\nthe original graph.\nAs the second step of the twist, the separation pairs are identified\nwith flipped orientation.\nIn figure~\\ref{fig:twists} the graphs are related by a twist around\nthe left-most and right-most nodes and thus have the same matroid.\n\nThese statements can be turned into an algorithm to detect shifts \\eqref{eq:shift_trafo}\nbetween vacuum sectors, which is sketched in the following and implemented in\n\\reduze.\nWe extend the graph isomorphism based shift detection by modifying the\ngenerated graphs with twists such that their canonical labels are minimized.\nA graph of a physical sector is decomposed into biconnected components with the\nalgorithm \\cite{HopcroftT71}.\nPossible separation pairs of biconnected components are identified via a\ndecomposition into \\emph{triconnected components}.\nWe implemented the algorithm \\cite{HopcroftT73, GutwengerMutzel2001}\nfor this purpose.\nIn order to generate at least one representative for each graph isomorphism\nclass, it is necessary to perform twists around separation pairs\nas specified by \\emph{virtual edges} as well as twists which correspond to\npermutations of edges within polygon components of the decomposition.\nWhile twisting we track the edges including their orientations to be able to\nidentify propagator momenta and different masses.\nGraphs with external legs are handled by intermediately joining their external\nnodes into one node and restricting to those twists, which keep the external\nlegs joined into one node with their original orientation.\n\nAlternatively, \\reduze\\ also offers a procedure to find all sector relations,\nwhich tries to identify sets of propagators in a straightforward approach\nbased on linear algebra and combinatorics.\nWhile this procedure finds all shifts between arbitrary sectors,\nincluding sectors not corresponding to a graph,\nit is usually much less efficient than the graph based methods.\n\n\n\\subsection{Sector symmetries}\n\\label{sec:sym_rel}\n\nSector symmetries are shifts of the form \\eqref{eq:shift_trafo} with\nmap a sector onto itself.\nAdditional permutations of the external momenta are permitted as long as the\nkinematic invariants are unchanged.\nDifferent such sector symmetries for physical sectors can be found by the\nunderlying symmetries of the graph as vertex permutations from the automorphism\ngroup and permutations of multi-edges. The automorphism group of a graph\nconsists of all permutations of the vertices which leave the canonical label\nof the graph unchanged. These transformations are calculated\nin \\reduze\\ with the algorithm \\cite{McKay1981} and are used to derive the\nmapping of the edges between pairs of vertices and the associated shift of\nthe loop momenta. In the case where there is more than one edge between\ntwo vertices also permutations of the edges with the same mass are considered.\n\nAlternatively, a complete set of sector symmetries can be calculated\nby \\reduze\\ using a combinatorial propagator matching approach.\n\n\n\\subsection{Matching of diagrams to sectors}\n\\label{sec:match_dias}\n\nAs discussed above, the assignment of momenta to propagators of Feynman\ndiagrams is ambiguous.\nIf diagrams are generated with a program like \\qgraf\\ \\cite{qgraf}, typically\nthe assigned loop momenta have to be shifted in order to index the involved\nloop integrals via integral families.\n\\reduze\\ can compute these shifts for diagram files generated with \\qgraf.\n\\reduze\\ can also handle permutations of external momenta and find matchings of\ndiagrams to crossed sectors.\n\n\n\\section{Distributed reduction algorithm}\n\n\\subsection{Load balanced system solving}\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}[b]{0.5\\textwidth}\n{\\small\n\\fcolorbox{framecolor}{shadecolor}{\n\\parbox{52mm}{\n\\mbox{}\\vspace*{-5mm}\n\\begin{align*}\n \\textcolor{emphcolor}{\\boldsymbol{I_5}} + c_{14} \\boldsymbol{I_4} + c_{13} \\boldsymbol{I_3} \\qquad\\quad\\,\\, \\qquad\\quad\\,\\,&= 0 \\\\\n \\textcolor{emphcolor}{\\boldsymbol{I_5}} + c_{24} \\boldsymbol{I_4} \\qquad\\quad\\,\\,  + c_{22} \\boldsymbol{I_2} \\qquad\\quad\\,\\,&= 0\\\\\n \\textcolor{emphcolor}{\\boldsymbol{I_5}} \\qquad\\quad\\,\\, + c_{33} \\boldsymbol{I_3} + c_{32} \\boldsymbol{I_2} \\qquad\\quad\\,\\,&= 0\n\\end{align*}\n\\mbox{}\\vspace*{-7mm}\n}}\n\\\\[1mm]\n\\fcolorbox{framecolor}{shadecolor}{\n\\parbox{58mm}{\n\\mbox{}\\vspace*{-5mm}\n\\begin{align*}\n\\qquad\\quad\\,\\,  \\textcolor{emphcolor}{\\boldsymbol{I_3}} + c_{42} \\boldsymbol{I_2} \\qquad\\quad\\,\\,  &= 0 \\\\\n\\qquad\\quad\\,\\,\\qquad\\quad\\,\\, \\textcolor{emphcolor}{\\boldsymbol{I_3}}\\qquad\\quad\\,\\,  + c_{51} \\boldsymbol{I_1} &= 0\n\\end{align*}\n\\mbox{}\\vspace*{-7mm}\n}}\n\\\\[1mm]\n\\fcolorbox{framecolor}{shadecolor}{\n\\parbox{58mm}{\n\\mbox{}\\vspace*{-5mm}\n\\begin{align*}\n\\qquad\\quad\\,\\,\\qquad\\quad\\,\\,\\qquad\\quad\\,\\, \\textcolor{emphcolor}{\\boldsymbol{I_2}} + c_{61} \\boldsymbol{I_1} &= 0\n\\end{align*}\n\\mbox{}\\vspace*{-7mm}\n}}\n}\\end{minipage}\n\\hspace*{5mm}\n\\includegraphics[width=0.35\\textwidth]{manager-worker-topo}\n\\end{center}\n\\caption{\\label{fig:blocks}\n\\emph{Left:} Three blocks of equations for loop integrals $I_i$\nwith coefficients $c_{ij}$ depending on kinematic invariants\nand the space--time dimension.\nEach block (shaded rectangle) contains equations\nfor the same leading integral (bold face $I_i$).\n\\emph{Right:} communication topology for MPI processes involved in a\ndistributed reduction.\n}\n\\end{figure}\n\n\\reduze\\ computes reductions for Feynman integrals by generating identities\nfor a range of seed integrals and reducing this linear system of equations.\nThe seed integrals are usually chosen for ranges in the propagator\nexponent sums $r$ and $s$.\nIn a typical application, reductions for integrals from several sectors are\nneeded.\nMoreover, a full reduction of a specific sector requires in general\nalso the reduction of subsector integrals.\n\\reduze\\  proceeds bottom-up: the reduction of a sector is started only after\nall subsector results are available.\nSectors which are no subsectors of each other can be reduced independently such\nthat these tasks are easily parallelized.\nThis kind of parallelization is available in the first version of \\reduze\\ \nvia a shell script, which launches programs for different sectors.\nIn version\\ 2 of \\reduze, two levels of parallelization are implemented via the\nmessage--passing--interface (MPI) standard.\nThe reduction of a sector becomes a \\emph{job} and\nseveral independent such reduction jobs can be executed in parallel.\nThe job system is described in more detail in the next section.\nOn top of this first level of parallelization, this version of \\reduze\\  implements\na distributed reduction algorithm for the reduction of a single sector.\nSince in this case the parallelization is less obvious we give some details about\nour implementation in the following subsection.\n\nIn \\reduze, a total ordering is defined for indexed integrals of the type\n\\eqref{eq:int_reduze_format}. The ordering defines integrals of a sector to be more\ncomplicated than integrals of its subsectors.\nIn the following, terms like ``leading'' or ``lower'' integral refer\nto this ordering.\nTo choose specific integrals as master\nintegrals the user may adjust the ordering, possibly at a later stage.\nIn order to reduce integrals of a given sector, IBP and LI identities are\ngenerated for a specified range of seed integrals.\nThis results in a sparse homogeneous linear system of equations for the indexed\nintegrals where the coefficients are rational functions of the kinematic\ninvariants and the space--time dimension.\nThe equations are sorted into blocks\nof equations with the same leading integral,\nsee left panel of figure \\ref{fig:blocks}.\n\nThe blocks are reduced bottom up, starting from the block with the lowest\nleading integral.\nFor each block, integrals are reduced, i.e. replaced by linear combinations of\nlower integrals, according to the results from lower blocks and subsectors\n(``back substitution'') if possible.\nIf a block contains several equations, one is kept and used to eliminate\nthe leading integral from all other equations in the block (``forward\nelimination''), which are subsequently solved for the new leading integral.\nThe coefficients of the integrals are normalized such that zeros are detected\nand numerator and denominator are coprime.\nThis requires multivariate polynomial greatest--common--divisor (GCD)\ncomputations which typically present the most time consuming part of the\nfull calculation.\nThe result of this reduction of a block is one equation for the block and\npossibly further equations with lower leading integrals to be inserted\ninto lower blocks.\nThe next block to be selected is the lowest block which contains more than\none equation or involves integrals which can be reduced.\n\n\\reduze\\  offers both, a purely serial reduction for one sector as well as\na distributed execution.\nIn the serial version, the above steps are performed in deterministic order\non one core.\nThe distributed version employs a star topology of MPI processes with one\nmanager and one or more workers, see right panel of figure~\\ref{fig:blocks}.\nThe workers perform the actual reduction steps for a block, while\nthe manager keeps track of the complete system of equations and balances the\nwork between the workers.\nMore specifically, an idle worker contacts the manager to request work.\nThe manager looks up the next block to process and sends its equations together\nwith equations needed for back substitutions to the worker.\nThe worker reduces the block and sends the result to the manager.\n\nOur motivation for this distributed algorithm is as follows.\nExperiments show that in typical applications the time needed for the reduction\nof one block can easily differ by more than 6 orders of magnitude.\nMoreover, the exact order of the individual reduction steps significantly\ndetermines the execution time for the full reduction and a bottom-up order\ntypically shows the best performance.\nBoth issues are directly addressed by the dynamical load balancing mechanism\npresented above, at least for a not--too--large number of worker processes.\nHow well this works in practice is quantified in the following subsection.\n\n\\reduze\\ allows to choose between \\ginac\\ \\cite{Bauer:2000cp}\nand \\fermat\\ \\cite{fermat} for the GCD computation\nneeded to normalize coefficients.\nDuring reduction, the equations are stored either in RAM or optionally\nin a transactional database as implemented by the open source Berkeley DB\n\\cite{berkeley}.\nWith transactions turned on, an aborted reduction of a single sector may be\nresumed at a later time; recovering completed reductions for sectors is\navailable in any case.\n\\reduze\\ supports different ways to split a calculation into several runs,\nthis is described in the tutorial provided with the \\reduze\\ distribution.\nReductions for crossed integrals are automatically be obtained via reduction\nresults from its uncrossed counterpart in order to save computation time\nand disk space.\n\n\\subsection{Performance results for single sectors}\n\\label{sec:perf_single}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.5\\textwidth]{perfsingle}}\n\\caption{\\label{fig:perf_single}\nExecution time for the reduction of single sectors in dependence\nof the number of worker processes employed for the reduction.\n}\n\\end{figure}\n\nPerformance results for the reduction of sectors for two--loop contributions\nto heavy--quark pair production are shown in figure~\\ref{fig:perf_single}.\nThese sectors have $t=4$ or $5$, respectively,\nand reductions were computed for integrals with $r=t\\ldots 7$ and\n$s=0\\ldots3$ or 4, respectively.\nWe used a computer with 48 CPU cores operating at $2.1$~GHz and started\n\\reduze\\ with $n_{\\text{workers}}+2$ MPI processes (one job center process\nand one manager process should be overbooked for a small number of available\ncores).\n\nIn general, we observe that the scaling with the number of processes is\nproblem specific and depends on the configuration of \\reduze, such as the\nchosen computer algebra system.\nThe upper two curves in the figure show an example with a good scaling for up\nto 22 workers.\nIndeed, we find examples where the scaling is good up to 48 workers.\nWe think this good scaling behavior is noteworthy, given the fact that it\ndescribes the distributed computation of a not too loosely coupled system.\nAs expected, we observe that beyond some number of worker\nprocesses the run time decreases less and less with additional workers\nand finally increases for even larger number of workers.\nContributions to this behavior is expected from serial parts\nof the code, communication overhead, but potentially also from\na ``less ideal'' order of evaluation when solving the system of equations\nwith a larger number of workers. \nThe lowest curve was obtained for an example of a reduction of a comparably\nsimple system of equations, where the onset of such a behavior is clearly\nvisible.\nIt is also not difficult to find examples with worse scaling,\nwhere a minimal runtime is obtained for only a few worker processes.\n\nUsing \\fermat\\ instead of \\ginac\\ for the GCD computations can easily result\nin a speed-up by an order of magnitude, see the two blue curves for\nsector~B in the figure.\nFor the displayed \\fermat\\ benchmarks, the system to be reduced was kept in RAM,\nwhile for the \\ginac\\ benchmarks a database was used.\nOur tests indicate that for examples of this type, the performance penalty for\nusing a database is considerably less than the differences due to the two\nalgebra systems considered here.\nNevertheless, also this option is relevant for optimal performance,\nespecially for a larger number of workers.\nTypically, if the coefficients in the equations become more involved, the\nreduction benefits considerably from a larger amount of workers compared\nto cases with compact coefficients.\nThe performance in a real application example, involving the reduction of\nseveral sectors, is discussed at the end of the next section.\n\n\n\\section{Job system}\n\n\\subsection{Load balanced execution of jobs}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.8\\textwidth]{jobcenter}}\n\\caption{\\label{fig:jobcenter}\nDynamical load balancing in \\reduze}\n\\end{figure}\n\nIn \\reduze, a \\emph{job} represents a sequence of computations which\ncan be performed once its dependencies, specified via the presence of input\nfiles, are fulfilled.\nMost jobs in \\reduze\\ are \\emph{serial jobs}: they are executed on a single\ncore but possibly in parallel to other jobs.\nParallel execution of different jobs represents the top layer parallelization\nmechanism of \\reduze\\ which is automatically available for any job type added\nto \\reduze.\nReduction of identities is an example for a \\emph{distributed job}.\nSuch a job can be executed by several processes in parallel: one process,\nthe \\emph{manager}, is responsible for the full execution of the job,\nother processes, the \\emph{workers}, help for some time with the execution.\nIn order to employ this second layer parallelization, a dedicated distributed\nalgorithm needs to be implemented for the job.\nFor each run of \\reduze, the user specifies a sequence of such jobs, which\nare inserted into a job queue. A job may generate additional auxiliary jobs\nautomatically. The job queue is responsible for resolving the dependencies\nbetween the jobs and determining the next job to be executed.\n\nIf \\reduze\\ is started with several MPI processes, one process will act as a\n\\emph{job center} and dynamically balance work between the other processes,\nthe \\emph{clients} of the job center, see figure~\\ref{fig:jobcenter}.\nThe job center schedules jobs using the job queue and assigns them to clients.\nAn idle client contacts the job center and requests work.\nThe job center responds by assigning a job to the client, either to be executed\nas a manager or as a worker.\nThe client changes its role accordingly.\nAs a worker, it contacts the responsible manager of the job and helps with the job\nexecution.\nAs a manager, it can either execute the job by itself or\nregister as an \\emph{employer} at the job center to request workers.\n\nIn order to optimize the efficiency of the parallelization,\nthe job center periodically collects performance measurements from the employers\nand estimates an optimal distribution of workers based on it.\nAccording to this estimate, workers will effectively be reassigned to other\nemployers by requesting release of workers from ``inefficient'' employers and\nassigning idle customer to ``efficient'' employers.\nThe basic idea is to avoid low efficiencies due to overloaded managers by\nassigning workers preferably to managers which idle a high percentage of their\nCPU time.\n\n\\subsection{Performance results for multiple sectors}\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}[b]{0.21\\textwidth}\n \\includegraphics[width=\\textwidth]{A_446}\\\\[2ex]\n \\includegraphics[width=\\textwidth]{A_247}\\\\[2ex]\n \\includegraphics[width=\\textwidth]{A_431}\\\\[2ex]\n \\includegraphics[width=\\textwidth]{A_191}\n\\vspace{2ex}\n\\end{minipage}\n\\hspace{10ex}\n\\includegraphics[width=0.5\\textwidth]{perfmulti}\n\\end{center}\n\\caption{\\label{fig:perf_multi}\nRuntime decrease for the reduction of multiple sectors using a large number\nof processor cores.\nThe reductions cover the sectors of the depicted graphs and all of their\nsubsectors.\nIn the graphs, thick lines represent massive propagators.\n}\n\\end{figure}\n\nPerformance results for the reduction of a selection of sectors are shown in\nfigure~\\ref{fig:perf_multi}.\nThese sectors are encountered in the calculation of two-loop corrections to\nheavy-quark pair production.\nReductions are calculated for integrals of the depicted sectors and all of\ntheir subsectors, where $r=t\\ldots 7$ and $s=0\\ldots 4$.\nThe benchmarks for the solid curve were performed on a computer with 48 CPU\ncores operating at $2.1$~GHz using \\fermat\\ and keeping all equations in RAM.\nThe measurements for the dashed curve were obtained using the ``Schr\\\"odinger''\ncluster of the University of Zurich with $2.8$~GHz cores; these runs were\nconfigured to use \\ginac\\ and a database.\n\nThe figure shows for this realistic application example that the calculation\nbenefits considerably from up to 96 cores, if available.\nLet us stress again, that the scaling behavior is problem specific and\nmay be worse for other types of applications.\nIn the present example,\nthe run--time decrease due to additional cores is quite close to an optimal\n$1\/n_{\\text{cores}}$ behavior for smaller numbers of cores.\n\n\n\\section{Other features}\n\n\\subsection{Differential equations for master integrals}\nOne method to solve master integrals consists\nof deriving differential equations by taking derivatives in the kinematic\ninvariants, replacing new integrals with the reduction results and solving\nthe differential equations by integration.\nIn particular for sectors with several independent integrals,\na change of basis for the master integrals may be required.\n\\reduze\\ offers the possibility to derive differential equations for Feynman\nintegrals and reduce these equations for some user choice of master integrals.\n\n\\subsection{Interference terms}\nStarting from diagrams generated by \\qgraf, \\reduze\\ can compute scalar\ninterferences of (bare) QED or QCD amplitudes in dimensional regularization.\nThis includes insertion of user--defined Feynman rules, contraction of Lorentz\nvector indices, performing Dirac traces, and evaluating color structures.\nIn case of an interference of a tree-level diagram with a diagram that\ncould be matched to a sector of an integral family (section \\ref{sec:match_dias})\nalso the occurring integrals, which belong to the matched sector and its subsectors,\nare indexed by the corresponding integral family.\nIn a further step, these integrals can be reduced to master integrals and\nsubstituted correspondingly in the interference terms.\nEach computation of an interference of two diagrams is treated as a job, and\nwhen MPI is used, these jobs can be performed in parallel.\nSpecific examples are distributed with the \\reduze\\ package.\n\n\n\\section{Usage}\n\nThe package \\reduze\\ can be downloaded from the web page\n \\url{http:\/\/projects.hepforge.org\/reduze}\\,.\nThe distribution contains a tutorial with detailed description of\ninstallation and usage as well as several example files.\n\n\n\\acknowledgments{\nWe thank York Schr\\\"oder for many useful discussions and comments on the\ndraft.\nWe are grateful to the CEDAR collaboration for hosting our project web page\non HepForge~\\cite{Buckley:2006np}.\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG SCHR 993\/2-1)\nand the Schweizer Nationalfonds (Grant 200020\\_124773\/1).}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}