diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjnhj" "b/data_all_eng_slimpj/shuffled/split2/finalzzjnhj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjnhj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\nIntegrable field theory in 1 dimension or on a whole line,\nboth at the classical\nand quantum levels, have been investigated quite intensively\nin recent years and\nmany interesting results have been uncovered. The motivation is mainly\ntwo-fold:\nfirstly, the integrable theory in its own right as a theoretical laboratory to\nstudy the structure of field theory beyond perturbation and secondly its\nconnection with conformal field theory (deformed CFT) and string theory.\n\nIn contrast, integrable field theory on a half line, say $x\\leq 0$,\n has a shorter history.\nHere, the effects of the boundary or the boundary conditions which replace\nthe ``asymptotic conditions\" in field theory on a whole line are the main\nobjects of research.\nThis problem is also related with integrable statistical lattice\nmodels with non-trivial boundary conditions, scattering of\nelectrons by an impurity in solids (the Kondo problem) monopole catalysed\nbaryon decays and\ndeformations of conformal field theory with boundaries.\nOne of the purposes of this paper is to discuss the effects caused by\nthe boundary on the integrable systems.\nContrary to the naive expectations, some ``integrable\" boundary interactions\ncan drastically change the character of the theory.\nIn some cases, for example, the sinh-Gordon theory,\n the theory becomes ill-defined because of the instability introduced by a\nspecific choice of ``integrable\" boundary interactions.\nThis will be discussed\nin some detail in later sections.\n\nThere are two types of approaches for the integrable field theory on a half\nline,\n algebraic and field theoretical. The algebraic approach was initiated by\nCherednik \\cite{Cher}\\thinspace more than ten years ago. Firstly, the dynamical system\nunder\nconsideration is integrable on the whole line having a factorisable S-matrix.\nSecondly, a natural assumption is that when restricted to the\nhalf line, {\\em the particle content (mass spectrum), and the S-matrices\ndescribing their mutual interactions, are exactly the same as those on\nthe whole line}.\nThirdly, when a particle hits the boundary it is assumed to be reflected\nelastically (up to rearrangements among mass degenerate particles).\nThe compatibility of the reflections and the scatterings constitutes the\nmain algebraic condition, called Reflection equation \\cite{Cher,GZa,Skl,GMN}\\thinspace\nwhich generalises\n the Yang-Baxter\nequation. In other words, the effect of the boundary is local and coded into\na set of reflection factors $K_{ab}(\\theta )$,\nwhere $a$ labels the incoming particle, $b$ labels the reflected particle and\n $\\theta$ is the rapidity.\n\nIn the case of integrable field theory with {\\em diagonal} S-matrices,\nfor example, affine Toda field theory to be discussed in detail in this paper,\nthe Yang-Baxter equation is trivially satisfied.\n The reflection at the boundary does not change the particle species and the\nReflection equation is again trivially satisfied. In this case a new algebraic\nequation called the Bootstrap equation governs the exact S-matrices and the\nReflection equation is replaced by Reflection Bootstrap equation \\cite{FK,RSb},\nwhich\nconstrains elastic reflection factors $K_a(\\theta )$.\n\nWhile the algebraic approach is applicable to any theories with\nexact S-matrices, the field theoretical approach is useful for theories having\nLagrangians and classical equations of motion.\nOur main concern in this paper is to discuss the boundary effects in the\nfield theoretic approach.\n Here the guiding principle is the\ninfinite set of conserved quantities which guarantees the integrability on a\nwhole line.\nThe boundary potential or the boundary interaction is so chosen as\nto preserve the set of conserved quantities or its suitable subset.\n Then the\nboundary effects can be deduced from the explicit forms of the boundary\ninteractions and field theoretical methods at the classical\nand\/or the quantum levels.\nFor well known integrable field theories on a whole line, sine-Gordon,\nnon-linear Schr\\\"odinger and affine Toda field theories, the ``integrable\"\nboundary interactions are deduced and\/or conjectured \\cite{Skl,Tar,CDRS,CDR}.\n It should be emphasised,\nhowever, that not all of these ``integrable\" boundary interactions give an\nintegrable field theory on a half line. Namely, preserving some infinite subset\nof conserved quantities is a necessary but not sufficient condition.\nCompatibility with the other principles of field theory, in particular, at the\nquantum level, must be checked\ncarefully.\n\nWe will discuss the boundary effects in integrable field theory on\na half line mostly taking explicit examples from affine Toda field theories.\nThis is because affine Toda field theory is one of the best understood\nintegrable field theory both at the classical and quantum levels.\nThe algebraic as well as the field theoretical approaches to affine Toda field\ntheory on a whole line have been very successful.\nThis paper is organised as follows:\nin section 2 affine Toda field theory is briefly reviewed in order to set the\nstage and to introduce notation. In section 3 very simple examples of a\nharmonic oscillator with a negative spring constant and a 1 dimensional wave\n(string) with ``integrable\" boundary interactions are discussed. It is shown\nthat the instability caused by a special ``integrable\" boundary interaction\nmakes the theory ill-defined.\n In section 4 the boundary effects in the simplest\nand best known affine Toda field theories, the sinh-Gordon and sine-Gordon\ntheories are discussed. Here it is shown that for a certain range of the\nparameter in the ``integrable\" boundary interaction the theory possesses\ninstability. In section 5 the boundary effects in affine Toda field theory,\nmainly the $a_n$ series are discussed. Section 6 is for summary and discussion.\n\n\n\n\\section{Affine Toda field theory on a half line}\n\\setcounter{equation}{0}\n\nAffine Toda field\ntheory \\cite{MOPa} is a\nmassive scalar field theory with exponential interactions in $1+1$\ndimensions\ndescribed by the action\n\\begin{equation}\nS=\\int dt\\int_{-\\infty}^\\infty\\thinspace dx{\\cal L}.\\label{wholeaction}\n\\end{equation}\nHere the Lagrangian density is given by\n\\begin{equation}\n{\\cal L}={1\\over 2}\n\\partial_\\mu\\phi^a\\partial^\\mu\\phi^a-V(\\phi )\\label{ltoda}\n\\end{equation}\nin which\n\\begin{equation}\nV(\\phi )={m^2\\over\n\\beta^2}\\sum_0^rn_ie^{\\beta\\alpha_i\\cdot\\phi}.\\label{vtoda}\n\\end{equation}\nHere $\\phi$ is an $r$-component scalar field, $r$ is the rank of a\ncompact semi-simple Lie algebra $g$ with $\\alpha_i$;\n$i=1,\\ldots,r$ being its simple roots. The roots are normalised so that long\nroots have length 2, $\\alpha_L^2=2$. An additional root,\n$\\alpha_0=-\\sum_1^rn_i\\alpha_i$ is an integer linear combination of the simple\nroots, is called the affine root;\n it corresponds to the extra spot on an extended Dynkin-Kac diagram\nfor $\\hat g$ and $n_0=1$.\nWhen the term containing the extra root is removed, the theory becomes\nconformally invariant (conformal Toda field theory).\nThe simplest affine Toda field theory, based on the simplest Lie algebra\n$a_1$, the algebra of $su(2)$, is called sinh-Gordon theory, a cousin\nof\nthe well known sine-Gordon theory. $m$ is a real parameter setting\nthe mass scale of the theory and $\\beta$ is a real coupling constant,\nwhich is relevant only in quantum theory.\n\n\nToda field theory is integrable at the classical level due to\nthe presence of an infinite number of conserved quantities.\nMany beautiful properties of Toda field theory, both at the classical\nand quantum levels, have been uncovered in recent years.\nIn particular, it is firmly believed that the integrability survives\nquantisation.\nThe exact quantum S-matrices are known\n\\cite{AFZa,BCDSa,BCDSc,CMa,DDa}\\cite{DGZc,CDS}\n for all the\nToda field theories based on non-simply laced algebras as well as those\nbased on simply laced algebras.\nThe singularity structure of the latter S-matrices, which in some\ncases contain poles up to 12-th order \\cite{BCDSc},\nis beautifully explained in terms of the\nsingularities of the corresponding Feynman diagrams\n\\cite{BCDSe}, so called Landau singularities.\n\n\n For the theory on a half line, \\rref{wholeaction} will be replaced by\n\\begin{equation}\nS=\\int dt\\left[\\int_{-\\infty}^0{\\cal L}dx -{\\cal\nB}\\right],\\label{halfaction}\n\\end{equation}\n where ${\\cal B}$, a function of the fields but not\ntheir derivatives, represents the boundary interaction.\nThe stationarity condition of the action implies the equation of motion\nwhich is the same as on the whole line and the boundary condition at\n$x=0$\n\\begin{equation}\n{\\partial\\phi\\over\\partial x}=-{\\partial{\\cal B}\n\\over \\partial\\phi}.\\label{todaboundary}\n\\end{equation}\nFor any choice of ${\\cal B}$, the energy\n\\begin{equation}\n{\\cal\nE}=\\int_{-\\infty}^0\\left[{1\\over2}(\\partial_t\\phi^a)^2+{1\\over2}(\\partial_x\\phi^a)^2+V(\\phi)\\right]dx+{\\cal B}\\label{Toda energy}\n\\end{equation}\nis always conserved. But it is {\\em no longer positive definite} for negative\nboundary interaction (${\\cal B}<0$).\n\nIn \\cite{CDRS,CDR}\\thinspace it was conjectured based on the analysis of low spin\nconserved quantities that the generic form of the\n``integrable\" boundary interaction is given by\n\\begin{equation}\n{\\cal B}={m\\over \\beta^2}\\sum_0^rA_ie^{{\\beta\\over 2}\n\\alpha_i\\cdot\\phi},\\label{allboundary}\n\\end{equation}\nwhere the coefficients $A_i,\\ i=0,\\dots ,r$ are a set of real numbers.\nThe condition\n\\rref{allboundary} is a generalisation of the well known results\nfor the sine-Gordon theory ($r=1$), in which case the coefficients\n$A_1$ and $A_0$ are completely arbitrary. However, for the affine Toda\nfield theory for higher rank algebras ($r\\ge2$),\nthe coefficients are severely constrained due to the presence of higher spin\nconserved quantities. For example, for the affine Toda field theories based\nupon\nthe\n$a_n^{(1)}$ series of Lie algebras the sequence of conserved charges includes\nall spins (except zero) modulo $n+1$.\n Except for $a_1^{(1)}$, which corresponds\nto the sinh-Gordon theory, each of these theories has conserved\ncharges of spin $\\pm 2$. It was shown in \\cite{CDRS} that\na combination of spin $\\pm 2$ conserved\ncharges as well as spin $\\pm 3$ conserved\ncharges are preserved in the\npresence of the\nboundary interaction if the boundary interaction term has the form\n\\rref{allboundary} with the further constraint:\n\\begin{equation}\n\\hbox{\\bf either}\\ A_i=\\pm2, \\ \\hbox{for}\\ i=0,\\dots ,n\\\n\\hbox{\\bf or}\\ A_i=0\\ \\hbox{for}\\ i=0,\\dots ,n\\ .\\label{anboundary}\n\\end{equation}\nBy similar analysis, a more general conjecture is obtained which applies to\nall simply-laced affine Toda field theories\n\\cite{CDR,BCDR}\\thinspace\n\\begin{equation}\n\\hbox{\\bf either}\\ A_i=\\pm2\\sqrt{n_i}, \\ \\hbox{for}\\ i=0,\\dots ,r\\\n\\hbox{\\bf or}\\ A_i=0\\ \\hbox{for}\\ i=0,\\dots ,r\\ .\\label{simplacedboundary}\n\\end{equation}\n\nThe sine- and sinh-Gordon theories and possibly the Bullough-Dodd theory\n(based on the $a_2^{(2)}$ algebra) and other non-simply laced theories\n\\cite{BCDR,AF}\nseem to be\nthe only ones for which there is a continuum of possible ``integrable\" boundary\ninteractions. We will show in section 4 that some part of the continuum might\nnot be realised in field theory. For the others, the possible ``integrable\"\nboundary interactions consist of a choice of signs.\n\n\n\\section{Simple examples of instability caused by boundary}\n\\setcounter{equation}{0}\n\n\nLet us start with a very simple example of a dynamical system with one degree\nof freedom, namely a harmonic oscillator with an arbitrary spring constant $k$,\nwhich is either {\\em positive} or {\\em negative}\n\\begin{equation}\nL={1\\over2}({dy(t)\\over dt})^2 -{1\\over2}k(y(t))^2.\\label{harmosci}\n\\end{equation}\nIn either case, the system has one conserved quantity, the energy\n\\begin{equation}\n{\\cal E}={1\\over2}({dy(t)\\over dt})^2 +{1\\over2}k(y(t))^2,\\label{harmener}\n\\end{equation}\nand satisfies the formal criterion of ``integrability\". The solutions of the\nequation of motion\n\\begin{equation}\n{d^2y\\over dt^2}=-ky\n\\end{equation}\nare oscillatory for {\\em positive} spring constant but they grow or decrease\nexponentially\nfor {\\em negative} spring constant $k$\n\\begin{equation}\ny(t)=\\pm e^{\\pm \\omega (t-t_0)}, \\quad \\omega^2=-k,\\quad \\hbox{for}\\ \\\nk<0;\\quad t_0\\ \\ \\hbox{arbitrary}.\\label{expsol}\n\\end{equation}\nNamely the system is unstable for negative $k$ and hardly qualifies to be\ncalled integrable in spite of the existence of the conserved energy.\nIn this case the energy is {\\em no longer positive definite} and fails to\nconstrain\nthe system. In fact it is easy to see that these unstable exponential\nsolutions have {\\em zero energy}.\nThe existence of non-trivial zero energy solutions (vacuum\nsolutions) would imply that the system can undergo certain changes without\ncosting energy, almost synonymous to instability.\nA {\\em negative energy solution} is also easily obtained\n\\begin{equation}\ny(t)=e^{x_0}\\cosh\\omega t, \\quad x_0\\ \\ \\hbox{arbitrary}.\\label{negosci}\n\\end{equation}\nThe energy can be made as large and negative as desired by choosing\nlarge $x_0$.\n A quantum version of such\nsystems, if any, would have serious difficulties.\n\nNext let us discuss the simplest integrable field theory on a half line, namely\na massless field (string) with a quadratic boundary potential,\n\\begin{equation}\n{\\cal L}={1\\over2}\\left[({\\partial\\phi(x,t)\\over{\\partial\nt}})^2-({\\partial\\phi(x,t)\\over{\\partial x}})^2\\right],\\quad {\\cal\nB}={1\\over2}A\\phi^2(0,t).\\label{quadbound}\n\\end{equation}\nThe equation of motion is the ordinary wave equation\n\\begin{equation}\n\\partial^2_t\\phi-\\partial^2_x\\phi=0\\label{waveeq}\n\\end{equation}\nand the boundary condition is\n\\begin{equation}\n{\\partial\\phi(x,t)\\over{\\partial x}}\\Bigm|_{x=0}=-A\\phi(0,t).\\label{linbound}\n\\end{equation}\nThis is a linear system,\nwith a conserved energy\n\\begin{equation}\n{\\cal E}=\n\\int_{-\\infty}^0\\left[{1\\over2}(\\partial_t\\phi)^2+{1\\over2}(\\partial_x\\phi)^2\\right]dx\n+{1\\over2}A\\phi^2(0,t).\\label{waveenergy}\n\\end{equation}\nTherefore it is ``integrable\" in the same sense as the harmonic oscillator.\n\nThe meaning of the boundary\ninteraction is now clear. It attaches a spring with spring constant $A$, which\nis positive or negative, to the endpoint of the string. If $A<0$ the energy is\n{\\em no\nlonger positive definite} and unstable solutions exist:\n\\begin{equation}\n\\phi(x,t)=\\pm e^{-A(x\\pm t-x_0)},\\quad \\hbox{for}\\ \\ A<0.\\label{waveunstab}\n\\end{equation}\nThey are finite everywhere on the half line $x\\leq0$ for finite time $t$ and\nlocalised near the boundary. It is again very easy to see that they have {\\em\nzero energy}. A {\\em negative energy solution} is easily obtained\n\\begin{equation}\n\\phi(x,t)=e^{-A(x-x_0)}\\cosh At , \\quad x_0\\\n\\hbox{arbitrary}.\\label{negstring}\n\\end{equation}\n\nIt is easy to find a `plane wave' basis satisfying the boundary condition\n\\begin{equation}\nu_p(x)\\propto (ip -A)e^{ipx}+(ip+A)e^{-ipx}.\\label{planebasis}\n\\end{equation}\nIt is also elementary to show that the above `plane wave' basis is orthogonal\nto the localised solution $e^{-Ax}$ for $A<0$. This means that the `plane wave'\nbasis is not complete for $A<0$. Therefore the initial value problem\n\\begin{equation}\n\\phi(x,t=0)=F(x),\\quad \\partial_t\\phi(x,t=0)=G(x), \\quad x\\leq0\\label{iniprob}\n\\end{equation}\nfor the\nstring on the half line with the boundary is {\\em unstable} for $A<0$ unless\n$F(x)$ and $G(x)$ are exactly orthogonal to\nthe localised solution $e^{-Ax}$. A quantum version of such a theory, if any,\nwould meet serious difficulties. This simple example shows clearly that some\nboundary effects, even if they are ``integrable\", are not ``local\" and can make\nthe theory ill-defined by the instability.\n\nOne can easily get the reflection factor \\cite{CDRS,GMN}\\ of the boundary from\nthe `plane wave' basis\n\\begin{equation}\nK(p)={ip+A\\over{ip-A}}\\label{simpleref}\n\\end{equation}\nfor either sign of $A$. But for $A<0$ such a result seems superficial because\nof the instability of the theory.\n\nIt is easy to note that adding a mass to the field tends to stabilise the\ntheory, as the mass term simply gives an attractive harmonic potential with a\nspring constant $m^2$ at each point:\n\\begin{equation}\n{\\cal\nL}={1\\over2}\\left[(\\partial_t\\phi(x,t))^2-(\\partial_x\\phi(x,t))^2-m^2\\phi(x,t)^2\\right],\\quad\n{\\cal B}={1\\over2}A\\phi^2(0,t).\\label{KGbound}\n\\end{equation}\nTherefore the {\\em zero energy} unstable mode exists only for $A\\leq-m$\n\\begin{equation}\n\\phi(x,t)=e^{\\pm\\omega t-A(x-x_0)},\\quad \\omega^2=A^2-m^2.\\label{KGzero}\n\\end{equation}\nThe latter condition is a disguise of the mass shell condition. A {\\em negative\nenergy solution} is given by\n\\begin{equation}\n\\phi(x,t)=\\cosh\\omega t\\thinspace e^{-A(x-x_0)},\n\\end{equation}\nand the energy is a function of $x_0$, ${\\cal E}={\\omega^2\\over{4A}}e^{2Ax_0}$,\nwhich can be as large and negative as $x_0\\to-\\infty$. The `plane wave' basis\nand\nthe reflection factor have the same form as the wave case.\n\n\n\\section{Sinh- and Sine-Gordon theories}\n\\setcounter{equation}{0}\n\nNext let us consider sinh-Gordon theory, the simplest member of the affine\nToda\nfield theories, based on the $a_1^{(1)}$ algebra. The Lagrangian density and\nthe boundary interactions in the notation of \\rref{ltoda}\\thinspace are\n\\begin{equation}\n{\\cal\nL}={1\\over2}\\left[(\\partial_t\\phi)^2-(\\partial_x\\phi)^2\\right]-{m^2\\over{\\beta^2}}\n\\left(e^{\\sqrt2\\beta\\phi}+e^{-\\sqrt2\\beta\\phi}\\right),\\ {\\cal\nB}={mA\\over{\\beta^2}}\n\\left(e^{\\beta\\phi\/\\sqrt2}+e^{-\\beta\\phi\/\\sqrt2}\\right).\\label{sinhlag}\n\\end{equation}\nHere the parameter $A$ is arbitrary. Since the exponential functions are always\npositive, it is expected that the boundary interaction term would induce strong\ninstability for large and negative $A$. We show this by constructing explicit\nclassical solutions for the equation of motion\n\\begin{equation}\n\\partial_t^2\\phi-\\partial_x^2\\phi=-2\\sqrt2{m^2\\over\\beta}\\sinh\\sqrt2\\beta\\phi,\n\\label{sinh-Gordoneq}\n\\end{equation}\nwith the boundary condition\n\\begin{equation}\n\\partial_x\\phi\\Bigm|_{x=0}=-\\sqrt2{mA\\over\\beta}\\sinh\\beta\\phi\/\\sqrt2\\Bigm|_{x=0}.\n\\label{sinhbc}\n\\end{equation}\nIt is elementary to see that for $A<0$\n\\begin{equation}\n\\tanh{\\sqrt2\\beta\\phi(x,t)\\over4}=e^{\\pm\\omega t}e^{-mA(x-x_0)},\\quad x_0>0\n\\label{sinhsol}\n\\end{equation}\nare solutions provided\n\\begin{equation}\n\\omega^2=m^2(A^2-4)\\geq0.\\label{sinhonshell}\n\\end{equation}\nFor the positive sign, the r.h.s. of \\rref{sinhsol}\\thinspace eventually exceeds 1,\nwhich is not allowed for tanh function with real arguments.\n\n\nNamely, for $A\\leq-2$ they are real and unstable solutions with one arbitrary\nparameter $x_0>0$. It is also elementary to check that they are {\\em zero\nenergy solutions}. For this the conserved energy takes the form\n\\begin{eqnarray}\n{\\cal\nE}&=&\\int_{-\\infty}^0\\left[{1\\over2}(\\partial_t\\phi)^2+\n{1\\over2}(\\partial_x\\phi)^2+{m^2\\over{\\beta^2}}\n\\left(e^{\\sqrt2\\beta\\phi}+e^{-\\sqrt2\\beta\\phi}-2\\right)\\right]dx\\nonumber\\\\\n& &+{mA\\over{\\beta^2}}\n\\left(e^{\\beta\\phi\/\\sqrt2}+e^{-\\beta\\phi\/\\sqrt2}-2\\right)\\Bigm|_{x=0},\n\\label{sinhenergy}\n\\end{eqnarray}\nin which constants are adjusted so that the trivial ``vacuum solution\" $\\phi=0$\nhas zero energy. This analysis also clarifies the dynamical meaning of the\n$A=-2$\ncondition \\rref{anboundary}\\thinspace, which is the critical point for the\ninstability.\nIt should be remarked that for $A=-2$ there exists a one-parameter ($x_0$)\nfamily of {\\em time independent zero energy solutions} which could be\ninterpreted\nas degenerate vacuua.\nThere are also {\\em negative energy solutions} for $A<-2$ . They are obtained\nby solving an initial value problem ($x_0>0$)\n\\begin{equation}\n\\phi(x,t=0)={4\\over{\\sqrt2\\beta}}\\hbox{arctanh} e^{-mA(x-x_0)},\\quad\n\\partial_t\\phi(x,t=0)=0.\\label{negsinh}\n\\end{equation}\nIt is easy to calculate the energy\n\\begin{equation}\n{\\cal E}={2m\\over{\\beta^2}}(A-{4\\over A}){a\\over{1-a}},\\quad\na=e^{2mAx_0},\\label{shboundless}\n\\end{equation}\nwhich goes to $-\\infty$ as $x_0\\to0$. In contrast, energy is bounded for\n$A\\geq-2$ \\cite{CDR}.\n\n Since sinh- and sine-Gordon theories are\nclosely related, it is expected that sine-Gordon theory with ``integrable\"\nboundary interaction has instability for certain range of the parameter $A$.\n The Lagrangian density and\nthe boundary interactions for sine-Gordon theory are\n\\begin{equation}\n{\\cal\nL}={1\\over2}\\left[(\\partial_t\\phi)^2-(\\partial_x\\phi)^2\\right]\n-{2m^2\\over{\\beta^2}}\\left(1-\\cos{\\sqrt2\\beta\\phi}\\right),\\ \\ {\\cal\nB}={2mA\\over{\\beta^2}}\\left(1-\\cos{\\beta\\phi\/\\sqrt2}\\right).\\label{sinelag}\n\\end{equation}\nHere the parameter $A$ is arbitrary. We have added constant terms to the\npotential and boundary interaction term in\nsuch a way that they vanish for the trivial ``vacuum solution\" $\\phi=0$. This\nalso\nmakes them positive definite (up to the sign of $A$). Since the cosine is a\nbounded function, it is expected that the\ninstability caused by the boundary interaction term, if any, would be milder\nthan\nthe sinh-Gordon case. We show this by constructing explicit classical solutions\nfor the equation of motion\n\\begin{equation}\n\\partial_t^2\\phi-\\partial_x^2\\phi=-2\\sqrt2{m^2\\over\\beta}\\sin\\sqrt2\\beta\\phi,\n\\label{sine-Gordoneq}\n\\end{equation}\nwith the boundary condition\n\\begin{equation}\n\\partial_x\\phi\\Bigm|_{x=0}=-\\sqrt2{mA\\over\\beta}\\sin\\beta\\phi\/\\sqrt2\\Bigm|_{x=0},\n\\label{sinebc}\n\\end{equation}\nin the same way as in the sinh-Gordon theory.\nIt is elementary to see that\n\\begin{equation}\n\\tan{\\sqrt2\\beta\\phi(x,t)\\over4}=e^{\\pm\\omega t}e^{-mA(x-x_0)},\\quad x_0\\ \\\n\\hbox{arbitrary}\n\\label{sinesol}\n\\end{equation}\nare solutions provided\n\\begin{equation}\n\\omega^2=m^2(A^2-4)\\geq0.\\label{sineonshell}\n\\end{equation}\n\nNamely, for $A^2\\geq4$ they are ordinary kink solutions with a specific $x$\ndependence. The one arbitrary\nparameter $x_0$ is interpreted as the kink position. In sharp contrast with the\nsinh-Gordon case, the field\n$\\phi$ is always finite for finite $x$ and $t$. However, for $A\\leq-2$\n(negative\nboundary term) they are {\\em zero energy\nsolutions}, a sign of instability. For this the conserved energy takes the\nform\n\\begin{eqnarray}\n{\\cal\nE}&=&\\int_{-\\infty}^0\\left[{1\\over2}(\\partial_t\\phi)^2+\n{1\\over2}(\\partial_x\\phi)^2+{2m^2\\over{\\beta^2}}\n\\left(1-\\cos{\\sqrt2\\beta\\phi}\\right)\\right]dx\\nonumber\\\\\n& &+{2mA\\over{\\beta^2}}\n\\left(1-\\cos{\\beta\\phi\/\\sqrt2}\\right)\\Bigm|_{x=0}\\geq{4mA\\over{\\beta^2}},\n\\label{sineenergy}\n\\end{eqnarray}\nin which constants are adjusted so that the trivial ``vacuum solution\" $\\phi=0$\nhas zero energy.\nAs in the case of sinh-Gordon theory, for $A=-2$ there exists a one-parameter\n($x_0$) family of {\\em time independent zero energy solutions} which could be\ninterpreted as degenerate vacuua. Negative energy solutions can also be\nobtained by solving an initial value problem similar to \\rref{negsinh}.\nHowever, in contrast with the sinh-Gordon theory, the energy is bounded from\nbelow.\n\nLet us conclude this section by remarking that field theoretical formulation of\nsinh- and\/or sine-Gordon theories with ``integrable\" boundary interaction for\n$A\\leq-2$ both at the classical and quantum levels seems problematic.\nIt might be possible to interpret the plethora of the solutions of Reflection\nBootstrap equation \\cite{RSb}\\thinspace in terms of the instability.\nFurther investigation is definitely wanted.\n\n\n\\section{$a_n^{(1)}$ Toda field theory}\n\\setcounter{equation}{0}\n\nAmong the $2^{n+1}$ possible choices of the ``integrable\" boundaries for\n$a_n^{(1)}$ Toda field theory \\rref{anboundary}, here we discuss only two\ncases,\nnamely all $A_i$ are the same\n\\begin{equation}\nA_i=A=\\pm 2,\\quad i=0,1,\\ldots,n.\\label{Znsymbc}\n\\end{equation}\nThe other cases, with mixed pluses and minuses, break the $Z_{n+1}$ symmetry\nwhich is essential for the determination of the bulk S-matrices\n\\cite{AFZa,BCDSc}. Moreover, in these cases, the ``vacuum\" $\\phi=0$ is no\nlonger\nthe solution of equation of motion and the boundary condition.\nTherefore, the ``new vacuum\", if any, of such theories is different from that\nof\nthe whole line.\nThe particle spectrum and their interactions are also different due to the lack\nof $Z_{n+1}$ symmetry and the different vacuum.\n\nFirst we show that $a_{2n+1}^{(1)}$ theory with $A=-2$ ``integrable\" boundary\ninteraction is {\\em unstable} in spite of the $Z_{2n+2}$ symmetry.\nWe have shown this for $a_1^{(1)}$ theory, the sinh-Gordon case.\n$a_{2n+1}^{(1)}$ theory with $A=-2$ has also a one parameter ($x_0$) family of\n{\\em zero energy solutions} (degenerate vacuua) of the same form as in the\nsinh-Gordon theory:\n\\begin{eqnarray}\n\\phi(x,t)&=&\\mu\\varphi(x,t),\\quad\n\\mu=\\alpha_1+\\alpha_3+\\cdots+\\alpha_{2n+1},\\nonumber\\\\\n\\tanh{\\beta\\varphi(x,t)\\over2}&=&e^{-A(x-x_0)}.\\label{anredsol}\n\\end{eqnarray}\nThe single component field $\\varphi$ in the special direction $\\mu$ satisfies\nall the $2n+1$ equations for $\\phi$. One only has to note $\\alpha_i\\cdot\\mu=2$\nfor $i$ odd and $-2$ for $i$\neven. This phenomenon is called ``dimension one\nreduction\" and has been discussed in some detail in \\cite{RSa}\\thinspace for various\ntypes of Toda field theory.\nHowever, it should be remarked that the reductions applicable to the whole line\nare not guaranteed to work for the boundary.\nThe existence of the zero (frequency) mode was also noted in \\cite{CDRS}\\thinspace\nwithin the linear approximation. It should be remarked that the above solution\ncan also be obtained by using Hirota's method.\n\nIn contrast, $a_{\\rm even}^{(1)}$ theory with $A=-2$ seems to have no\ndegenerate vacuua.\nSo it deserves further investigation. Let us consider the weak coupling limit\n($\\beta\\ll1$), or the linear approximation.\nThen the system is a sum of independent Klein-Gordon fields, with ``integrable\"\nquadratic boundary interactions:\n\\begin{eqnarray}\n{\\cal L}&=&\\sum_{j=1}^{2n}|\\partial_\\mu\\phi_j|^2-m_j^2|\\phi_j|^2,\\quad\nm_j=2m\\sin{j\\pi\\over{2n+1}},\\nonumber\\\\\n{\\cal B}&=&-{1\\over{2m}}\\sum_{j=1}^{2n}m_j^2|\\phi_j|^2.\\label{linappr}\n\\end{eqnarray}\nDue to the attractive boundary effect, each field has a localised solution,\n$e^{{m_j^2\\over{2m}}x}$, namely a boundary bound state.\nAs before this state is orthogonal to the `plane wave' basis\n$$\nu_p^{(j)}(x)\\propto (ip\n+{m_j^2\\over{2m}})e^{ipx}+(ip-{m_j^2\\over{2m}})e^{-ipx}.\n$$\nThis means that upon quantisation, one needs to introduce the creation and\nannihilation operators for the `plane wave' states as well as the boundary\nbound states:\n\\begin{eqnarray}\n\\phi_j(x,t)&=&{1\\over{2\\pi}}\\int_0^\\infty{dp\\over\\sqrt{N_p}}\\left\\{e^{-i\\omega_pt}a_j(p)+e^{i\\omega_pt}a_{\\bar\nj}^\\dagger(p)\\right\\}u_p^{(j)}(x) \\nonumber\\\\\n&+&{1\\over\\sqrt{N_j}}\\left\\{e^{-i\\omega_jt}b_j+e^{i\\omega_jt}b_{\\bar\nj}^\\dagger\\right\\}e^{{m_j^2\\over{2m}}x}.\\label{quantise}\n\\end{eqnarray}\nHere $N_p$ and $N_j$ are normalisation constants and $\\omega_p^2=p^2+m_j^2$,\n$\\omega_j^2=m_j^2(1-{m_j^2\\over{4m^2}})$.\nOtherwise the Heisenberg commutation relations\n$$\n[\\phi_j(x,t),\\partial_t\\phi_k(y,t)]=i\\delta_{j\\bar k}\\delta(x-y)\n$$\ncannot be satisfied. Namely the spectrum is changed by the boundary.\nTherefore a naive correspondence with the algebraic approach \\`a la Cherednik\nseems to be lost. However, it is tempting to relate the emergence of the\nboundary bound states with the plethora of the solutions\nof the Reflection Bootstrap equation.\nCertain characteristic differences between the solutions of reflection\nBootstrap equation for $a_{\\rm even}$ and $a_{\\rm odd}$ theories are also\nreported \\cite{RSb}.\n\n\\section{Summary and discussion}\n\\setcounter{equation}{0}\n\nThe effects of the `boundary interactions' are analysed for various types of\nintegrable field theories on a half line.\nHere we discuss only the ``integrable'' boundary interactions such that they\npreserve certain subset of the classical infinite set of conserved quantities.\nIt is shown that not all of these `integrable' boundary interactions give\nconsistent quantum field theories on a half line.\nEspecially when the boundary interaction is negative and strong, some theories\nbecome ill-defined due to the instabilities which are related with the\nnon-positive definite energy.\n\nFor sinh-Gordon theory with a strong negative boundary interaction,\nan explicit 1-parameter ($x_0$) family of unstable solutions \\rref{sinhsol}\\thinspace\nare constructed, which has the form of 1-soliton solution with\n$x_0$ being its position.\nIt is well known that sinh-Gordon theory and other affine Toda field theories\non a whole line have no soliton solutions. The negative boundary interaction\nenables the soliton solution and thereby causes the instability.\nIt is also interesting that the unstable solution can be obtained from the\n``trivial vacuum\" $\\phi=0$ by the B\\\"acklund Transformation (B-T).\nThe close similarity between the B-T and the ``integrable\" boundary\ninteractions \\cite{Tar,SSW} will be discussed elsewhere \\cite{FSb}.\n\nAlthough we did not discuss the free boundary, ${\\cal B}=0$, in this paper,\nthis does not mean the free boundary is uninteresting.\nThe free boundary always preserves the necessary conserved quantities and\ntherefore is ``integrable\" ,\nnot only for affine Toda field theory but also for other types of theories\nlike non-linear sigma models which have non-diagonal S-matrices.\nThe corresponding Neumann boundary condition, $\\partial_x\\phi=0$ at $x=0$\nis always satisfied if $\\phi$ is extended to the whole line as an even\nfunction.\nIt is an interesting challenge to derive reflection factors $K_a(\\theta)$\ncorresponding to the free boundary for various models in terms of the field\ntheoretical methods.\n\n\n\n\n\\section*{Acknowledgments}\nWe thank Ed Corrigan and Q-H.\\thinspace Park for fruitful discussion. R.\\thinspace S is\ngrateful for Japan Society for Promotion of Sciences for financial support\nwhich enabled him to visit UK.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction }\n \\label{sec-introd}\n \\andy{intro}\n\nThe classical and quantum dynamics of bound Hamiltonian systems\nunder the action of periodic ``kicks\" are in general very\ndifferent. Classical systems can follow very complicated\ntrajectories in phase space, while the evolution of the wave\nfunction in the quantum case is more regular. In the classical\ncase, in those regions of the phase space that are stochastic, the\nevolution of the system can be well described in terms of the\naction variable alone and one of the most distinctive features of\nan underlying chaotic behavior is just the diffusion of the action\nvariable in phase space. On the other hand, in the quantum case,\nsuch a diffusion is always suppressed after a sufficiently long\ntime \\cite{Chirikov,Berry}. This phenomenon, known as the quantum\nmechanical suppression of classical chaos, can be framed in a\nproper context in terms of the semiclassical approximation $\\hbar\n\\to 0$ \\cite{CCSG,Tabor}.\n\nThe ``kicked\" rotator is a pendulum that evolves under the action\nof a gravitational field that is ``switched on\" at periodic time\nintervals. It is a very useful system, able to elucidate many\ndifferent features between the classical and the quantum case. By\nstudying this model, Kaulakis and Gontis \\cite{KaulGontis} showed\nthat a diffusive behavior of the action variable takes place even\nin the quantum case, if a quantum measurement is performed after\nevery kick. This interesting observation was investigated in some\ndetail in a recent paper \\cite{FPS}, where it was proven that\nquantum mechanical measurements of the action variable provoke\ndiffusion in a very large class of ``kicked\" systems, even when the\ncorresponding classical dynamics is regular. In this paper we shall\nfirst briefly review some of our general results and then\ncorroborate our findings by concentrating our attention on the\nparticular case of the kicked rotator.\n\n\n\\setcounter{equation}{0}\n\\section{Kicks interspersed with quantum measurements }\n \\label{sec-kickmeas}\n \\andy{kickmeas}\n\n\nWe consider the Hamiltonian\n\\andy{eq:hamilt}\n\\begin{equation}\nH=H_0(p) + \\lambda V(x) \\delta_T(t),\n\\label{eq:hamilt}\n\\end{equation}\nwhere $p$ and $x \\in [-\\pi,\\pi]$ are the action and angle variable,\nrespectively, and\n\\andy{eq:deltat}\n\\begin{equation}\n\\delta_T(t)=\\sum_{k=-\\infty}^{\\infty} \\delta (t-kT),\n\\label{eq:deltat}\n\\end{equation}\n$T$ being the period of the perturbation. We impose periodic\nboundary conditions on the interaction $V(x)$. This Hamiltonian\ngives rise to the so-called radial twisting map, that describes the\nlocal behavior of a perturbed integrable map near resonance\n\\cite{Lichten}. The free Hamiltonian $H_0$ has a discrete spectrum\nand a countable complete set of eigenstates $\\{|m\\rangle\\}$:\n\\andy{eigenfun}\n\\begin{equation}\n\\langle x|m\\rangle=\\frac{1}{\\sqrt{2\\pi}} \\exp\\left(imx\\right),\n\\qquad m=0,\\pm1,\\pm2,\\ldots\\ .\n\\label{eq:eigenfun}\n\\end{equation}\nWe shall consider the evolution engendered by the Hamiltonian\n(\\ref{eq:hamilt}) interspersed with quantum measurements, in the\nfollowing sense: the system evolves under the action of the free\nHamiltonian for $(N-1)T+\\tau \\lambda_{\\rm crit}$ \\\\\n \\hline\nB &quantum & no diffusion \\\\\n \\hline\nC &quantum + measurements & diffusion $\\forall \\lambda$ \\\\\n \\hline\nD &classical + random & diffusion $\\forall \\lambda$ \\\\\n \\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\n\\setcounter{equation}{0}\n\\section{Semiclassical limit}\n \\label{sec-semicl}\n \\andy{semicl}\n\nAs we have seen, the effect of quantum measurements is basically\nequivalent to a complete randomization of the classical angle\nvariable $x$, at least if one's attention is limited to the\ncalculation of the diffusion coefficient in the chaotic regime. One\nmight therefore naively think that the randomized classical map\n(\\ref{eq:classmeas}) and the quantum map with measurements\n(\\ref{eq:master}), (\\ref{eq:dhzero0})-(\\ref{eq:diffusion}) are\nidentical. This expectation would be wrong: there are in fact\ncorrections in $\\hbar$. It is indeed straightforward, using\nEqs.~(\\ref{eq:energy2})-(\\ref{eq:baklemma}), to obtain in the\nquantum case\n\\begin{eqnarray}\n\\langle p^3\\rangle_N &=& \\langle p^3\\rangle_{N-1}+3\\lambda^2\n\\langle f^2\\rangle\\langle p\\rangle_{N-1}+\\lambda^3\\langle f^3\\rangle,\n\\nonumber\\\\\n\\langle p^4\\rangle_N &=& \\langle p^4\\rangle_{N-1}+6\\lambda^2\n\\langle f^2\\rangle\\langle p^2\\rangle_{N-1}\n+4\\lambda^3\\langle f^3\\rangle\\langle p\\rangle_{N-1}\n+\\lambda^4\\langle f^4\\rangle +\\lambda^2\\hbar^2\\langle\n(f')^2\\rangle. \\nonumber \\\\\n\\end{eqnarray}\nOn the other hand, using (\\ref{eq:classmeas}) and the periodic\nboundary conditions, one gets for the randomized classical map\n\\begin{eqnarray}\n\\langle\\langle p_N^3\\rangle\\rangle\n&=& \\langle\\langle p_{N-1}^3\\rangle\\rangle+3\\lambda^2\n\\langle f^2\\rangle\\langle\\langle p_{N-1}\\rangle\\rangle\n+\\lambda^3\\langle f^3\\rangle,\n\\nonumber\\\\\n\\langle\\langle p_N^4\\rangle\\rangle\n&=& \\langle\\langle p_{N-1}^4\\rangle\\rangle+6\\lambda^2\n\\langle f^2\\rangle\\langle\\langle p_{N-1}^2\\rangle\\rangle\n+4\\lambda^3\\langle f^3\\rangle\\langle\\langle p_{N-1}\\rangle\\rangle\n+\\lambda^4\\langle f^4\\rangle.\n\\end{eqnarray}\nHence the two maps have equal moments up to third order, while the\nfourth moment displays a difference of order $O(\\hbar^2)$:\n\\andy{fourth}\n\\begin{equation}\n\\langle p^4 \\rangle_N - \\langle p^4 \\rangle_{N-1}\n=\\langle\\langle p_N^4\\rangle\\rangle\n-\\langle\\langle p_{N-1}^4\\rangle\\rangle\n+\\lambda^2\\hbar^2\\langle (f')^2\\rangle.\n\\label{eq:fourth}\n\\end{equation}\n\nIn order to understand better the similarities and differences\nbetween the two maps, as well as the quantum mechanical\ncorrections, we focus our attention on the particular case of the\nkicked rotator $H_0=p^2\/2$, $V(x)=\\cos x$, which gives rise to the\nso-called standard map\n\\andy{eq:standmap}\n\\begin{eqnarray}\nx_{N} &=& x_{N-1}+p_{N-1}T, \\nonumber \\\\\np_{N} &=& p_{N-1} +\\lambda \\sin x_{N}.\n\\label{eq:standmap}\n\\end{eqnarray}\nThe conditional probability density $W_{\\rm cl}$ that an initial\nstate $(p',x')$ evolves after one step into the final state\n$(p,x)$ is, from (\\ref{eq:standmap}),\n\\begin{eqnarray}\nW_{\\rm cl}(p,x|p',x')&=&\\delta(p-p'-\\lambda\\sin\nx)\\;\\delta(x-x'-p'T)\\nonumber\\\\\n&=&\\delta(p-p'-\\lambda\\sin[x'+p'T])\\;\\delta(x-x'-p'T).\n\\end{eqnarray}\nThis is a completely deterministic evolution. On the other hand, if\none randomizes the standard map, as in (\\ref{eq:classmeas}),\n\\andy{eq:randstandmap}\n\\begin{eqnarray}\nx_{N} &=& \\xi_{N}, \\nonumber \\\\\np_{N} &=& p_{N-1} +\\lambda \\sin x_{N},\n\\label{eq:randstandmap}\n\\end{eqnarray}\nthe conditional probability density becomes\n\\andy{cltransprobtot}\n\\begin{eqnarray}\nW_{\\rm cl}(p,x|p',x')=W_{\\rm\ncl}(p,x|p')=P(x)\\;\\delta(p-p'-\\lambda\\sin x)\n=\\frac{1}{2\\pi}\\delta(p-p'-\\lambda\\sin x)\n\\label{eq:cltransprobtot}\n\\end{eqnarray}\nand is independent of the initial position $x'$. It is therefore\npossible to describe the dynamics by considering only the momentum\ndistribution\n\\andy{cltransprob}\n\\begin{eqnarray}\nW_{\\rm cl}(p|p') &=&\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi\ndx\\;\\delta(p-p'-\\lambda\\sin x)\n=\\frac{1}{\\lambda\\pi}\\int_{-1}^{+1}\\frac{dy}{\\sqrt{1-y^2}}\\;\n\\delta\\left(y-\\frac{p-p'}{\\lambda}\\right)\n\\nonumber\\\\\n&=&\\frac{1}{\\pi}\\frac{1}{\\sqrt{\\lambda^2-(p-p')^2}}\\;\\theta(\\lambda-|p-p'|).\n\\label{eq:cltransprob}\n\\end{eqnarray}\nNotice that $W_{\\rm cl}(p|p')$ is a function of the momentum\ntransfer $|\\Delta p|=|p-p'|$ and vanishes for $|\\Delta p|>\\lambda$.\n\nConsider now the kicked quantum rotator with measurements. From\nEq.\\ (\\ref{eq:transprob}), the transition probability reads\n\\andy{qutransprobdef}\n\\begin{equation}\nW_{\\rm q}(p=\\hbar n|p'=\\hbar n')=\\frac{1}{\\hbar}\nW_{nn'}=\\frac{1}{\\hbar}\\left|\\langle n|e^{-i\\lambda \\cos x\n\/\\hbar}|n'\\rangle\\right|^2\n\\label{eq:qutransprobdef}\n\\end{equation}\nand by using the definition (\\ref{eq:eigenfun}) one obtains\n\\andy{transbessel}\n\\begin{eqnarray}\n\\langle n|e^{-i\\lambda \\cos x\/\\hbar}|n'\\rangle&=&\n\\int_{-\\pi}^{\\pi} dx \\langle n|x\\rangle\ne^{-i\\lambda \\cos x\/\\hbar} \\langle x|n'\\rangle\n\\nonumber\\\\\n&=&\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} dx \\;e^{-i(n-n')x}\ne^{-i\\lambda \\cos x\/\\hbar}=i^{n-n'}\nJ_{n-n'}\\left(\\frac{\\lambda}{\\hbar}\\right),\n\\label{eq:transbessel}\n\\end{eqnarray}\nwhere $J_{m}(z)$ is the Bessel function of order $m$. Therefore, in\nthe quantum case, from (\\ref{eq:qutransprobdef}) and\n(\\ref{eq:transbessel}), we can write\n\\andy{qutransprob}\n\\begin{equation}\nW_{\\rm q}(p=\\hbar n|p'=\\hbar n')=\\frac{1}{\\hbar}\nJ_{\\nu}\\left(\\frac{\\lambda}{\\hbar}\\right)^2 \\qquad (\\Delta\np=p-p'=\\hbar \\nu; \\;\\; \\nu \\equiv n-n').\n\\label{eq:qutransprob}\n\\end{equation}\nThere are two important differences between the classical case\n(\\ref{eq:cltransprob}) and its quantum counterpart\n(\\ref{eq:qutransprob}): i) the quantum mechanical transition\nprobability $W_{\\rm q}$ admits only quantized values of momentum\n$\\hbar n$, while the classical one $W_{\\rm cl}$ is defined on the\nreal line; ii) momentum can change by any value in the quantum case\n(notice however that this occurs with very small probability for\n$|\\Delta p|=\\hbar |\\nu|\\gg\\lambda$ \\cite{Chirikov}), while in the\nclassical case this change is strictly constrained by $|\\Delta\np|\\leq\\lambda$. These features have an interesting physical\nmeaning: see Figure 1. The transition probability of classical\nmomentum appears as an ``average\" of its quantum counterpart, which\nexplains the strong analogy discussed in Section 2. At the same\ntime, the quantum mechanical transition probability has a small\nnonvanishing tail for $|\\Delta p|=\\hbar |\\nu|>\\lambda$: this is at\nthe origin of the difference (\\ref{eq:fourth}).\n\\begin{figure}[t]\n\\epsfig{file=bessel.eps, width=\\textwidth}\n\\caption{Momentum transition probabilities for the kicked rotator\n($\\lambda = 100\\hbar$ and the momentum transfer $p-p'$ is expressed\nin units $\\hbar$). The thick line is the classical expression\n(\\ref{eq:cltransprob}): it diverges for $p-p'=\\lambda$ and vanishes\nfor $p-p' > \\lambda$. The quantum mechanical transition probability\n(\\ref{eq:qutransprob}) is defined only for integer values of $p-p'$\n(dots). The interpolating line (obtained by treating the order of\nthe Bessel function as a continuos variable) oscillates around its\nclassical counterpart and is nonvanishing (although very small)\noutside the classical range, i.e.\\ for $p-p' > \\lambda$. }\n\\label{fig:bessel}\n\\end{figure}\n\nFinally, let us show how one recovers the transition probability\n$W_{\\rm cl}$ starting from $W_{\\rm q}$, in the semiclassical limit.\nWe look at the limit $\\hbar\\to0$, while keeping $\\Delta p=\\hbar\\nu$\nfinite:\n\\andy{limit}\n\\begin{equation}\n\\hbar\\to0,\\quad\\nu\\to\\infty\\quad\\mbox{with}\\quad\\Delta p=\\hbar\\nu=\\mbox{const}.\n\\label{eq:limit}\n\\end{equation}\nIn this limit, the argument and the order of the Bessel function in\n(\\ref{eq:qutransprob}) are infinities of the same order. For\n$|\\Delta p|\\leq\\lambda$, setting $\\Delta p\/\\lambda\n\\equiv\\cos\\beta$, one gets\n\\begin{equation}\n\\frac{\\lambda}{\\hbar}=\\frac{\\lambda}{\\Delta p}\\frac{\\Delta p}{\\hbar}\n=\\nu \\sec\\beta.\n\\end{equation}\nHence, by using the asymptotic limit of the Bessel function \\cite{tables}\n\\andy{asympt}\n\\begin{equation}\nJ_{\\nu}(\\nu\\sec\\beta)\n\\stackrel{\\nu\\; {\\rm large}}{\\sim}\n\\sqrt{\\frac{2}{\\nu\\pi\\tan\\beta}}\n\\left[\\cos\\left(\\nu\\tan\\beta-\\nu\\beta-\\frac{\\pi}{4}\\right)\n+{\\rm O}(\\nu^{-1})\\right],\n\\label{asympt}\n\\end{equation}\nEq.~(\\ref{eq:qutransprob}) becomes, in the limit (\\ref{eq:limit}),\n\\andy{semicltransprob}\n\\begin{eqnarray}\n& &W_{\\rm q}(p|p')=\\frac{1}{\\hbar} J_{\\frac{\\Delta\np}{\\hbar}}\\left(\\frac{\\lambda}{\\hbar}\\right)^2=\\frac{1}{\\hbar}\nJ_{\\nu}\\left(\\nu\\sec\\beta\\right)^2\\nonumber\\\\ & &\\sim\n\\frac{1}{\\hbar}\\frac{2}\n{\\frac{\\Delta p}{\\hbar}\\pi\\sqrt{\\frac{\\lambda^2}{\\Delta p^2}-1}}\n\\left[\n\\cos^2\\left(\\frac{\\Delta p}{\\hbar}\\sqrt{\\frac{\\lambda^2}{\\Delta p^2}-1}\n-\\frac{\\Delta p}{\\hbar}\\arccos\\frac{\\Delta p}{\\lambda}-\\frac{\\pi}{4}\\right)\n+{\\rm O}\\left(\\frac{\\hbar}{\\Delta p}\\right)\\right]\n\\nonumber\\\\ & &\\sim\nW_{\\rm cl}(p|p')\\left[1+\n\\sin\\left(\\frac{2\\sqrt{\\lambda^2-\\Delta p^2}}{\\hbar}\n-\\frac{2\\Delta p}{\\hbar}\\arccos\\frac{\\Delta p}{\\lambda}\\right)\n+{\\rm O}\\left(\\frac{\\hbar}{\\Delta p}\\right)\\right],\n\\nonumber \\\\\n& & \\hspace{8cm} (|\\Delta p|\\leq\\lambda)\n\\label{eq:semicltransprob}\n\\end{eqnarray}\nthat, due to Riemann-Lebesgue lemma, tends to $W_{\\rm cl}$ in the\nsense of distributions.\n\nOn the other hand, for $|\\Delta p|>\\lambda$, setting $\\Delta\np\/\\lambda \\equiv \\cosh\\alpha$ and using the asymptotic formula \\cite{tables}\n\\andy{asympt1}\n\\begin{equation}\nJ_{\\nu}\\left(\\frac{\\nu}{\\cosh\\alpha}\\right)\\stackrel{\\nu\\;\n{\\rm large}}{\\sim}\n\\frac{\\exp(\\nu\\tanh\\alpha-\\nu\\alpha)}{\\sqrt{2\\nu\\pi\\tan\\beta}}\n\\left[1+{\\rm O}(\\nu^{-1})\\right],\n\\label{asympt1}\n\\end{equation}\nwe get\n\\andy{semicltransprob1}\n\\begin{eqnarray}\n& &W_{\\rm q}(p|p')\\nonumber\\\\ & &\\sim\n\\frac{1}{2\\pi\\sqrt{\\Delta p^2-\\lambda^2}}\n\\exp\\left\\{-\\frac{2\\Delta p}{\\hbar}\n\\left[\\arccos\\frac{\\Delta p}{\\lambda}\n-\\sqrt{1-\\left(\\frac{\\lambda}{\\Delta p}\\right)^2}\\right]\n\\right\\}\\left[1+{\\rm O}\\left(\\frac{\\hbar}{\\Delta p}\\right)\\right],\n\\nonumber \\\\\n& & \\hspace{8cm} (|\\Delta p|>\\lambda)\n\\label{eq:semicltransprob1}\n\\end{eqnarray}\nwhich vanishes exponentially (remember that $\\tanh\\alpha<\\alpha$).\nEquations (\\ref{eq:semicltransprob}) and\n(\\ref{eq:semicltransprob1}) corroborate the results of Section 2\nand enable us to conclude that the ``randomized\" classical kicked\nrotator is just the semiclassical limit of the ``measured\" quantum\nkicked rotator.\n\n\\setcounter{equation}{0}\n\\section{Concluding remarks }\n \\label{sec-conclrem}\n \\andy{conclrem}\nThe conclusion drawn in the previous section for the kicked rotator\ncan be generalized to an arbitrary radial twisting map. The\ncalculation and the techniques utilized are more involved and will\nbe presented elsewhere. There are also a number of related problems\nthat deserve attention and a careful investigation. Among these, we\njust mention the case of imperfect quantum measurements, yielding a\npartial loss of quantum mechanical coherence, the relation to\ndisordered systems, Anderson localization \\cite{Flores} and quantum\nZeno effect \\cite{QZE} and finally the extension to a different\nclass of Hamiltonians \\cite{Casati}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Contribution}\n\n\\subsection{Multi-player Bandits}\nWe consider the multi-player multi-armed bandit problem with $K$ arms and $M < K$ players. At each time $t=1,...,T$, where $T$ is the time horizon, each player $m=1,...,M$ selects an arm denoted by $\\pi_m(t) \\in \\{1,...,K\\}$. If arm $k=1,...,K$ has been selected by strictly more than one player we say that a collision occurs on arm $k$, and we denote by \n$\n\\eta_{k}(t) = \\mathbbm{1} \\left\\{ \\sum_{m=1}^{M} \\mathbbm{1}\\{ \\pi_m(t) = k \\} \\ge 2 \\right\\}\n$\nthe corresponding indicator function. The reward received by player $m=1,...,M$ is denoted by\n$\n r_m(t) = X_{\\pi_m(t)}(t) [1-\\eta_{\\pi_m(t)}(t)]\n$\nwhere $(X_k(t))_{t \\ge 1}$ is i.i.d. Bernoulli with mean $\\mu_k$ for $k=1,...,K$. In other words, if a player $m$ is involved in a collision then he receives a reward of $0$, otherwise he receives a binary reward with mean $\\mu_{\\pi_m(t)}$.\n\n\\subsection{No Collision Information}\n\nWe consider the case where each player $m=1,...,M$ only observes its own reward $r_m(t)$ and nothing else. He does not observe the rewards obtained by other players, he does not know which arms have been selected by other players, and he cannot observe the collision indicator $\\eta_{\\pi_m(t)}$. This case is called \"without collision sensing\", and is the most arduous because when player $m$ gets a reward of $r_m(t) = 0$, he does not know whether this was caused by a collision so that $\\eta_{\\pi_m(t)}(t) = 1$ or whether this was caused because the reward of the chosen arm was null i.e. $X_{\\pi_m(t)}(t) = 0$. Of course, when a reward of $r_m(t) = 1$ is obtained, he can however be sure that no collision has occurred. Therefore, the problem is fully decentralized, and no communication is allowed between the players.\n\n\\subsection{Regret Minimization}\n\n Recall that $\\mu_1,...,\\mu_K$ denotes the mean reward of each arm where no collision has occurred, and define $\\mu_{(1)},....,\\mu_{(K)}$ its version sorted in descending order, so that $\\mu_{(1)} \\ge \\mu_{(2)} \\ge ... \\ge \\mu_{(K)}.$ The goal for the players is to maximize the expected sum of rewards, and to do so, each player must select a distinct arm amongst the $M$ best arms to both avoid any collisions and maximize total reward. The regret (sometimes called pseudo-regret) is defined as the difference between the total expected reward obtained by an oracle that plays the optimal decision and the reward obtained by the players\n$$\n R(T,\\mu) = \\sum_{t=1}^{T} \\sum_{m=1}^{M} \\mu_{(m)} - \\mathbb{E}( r_{m}(t) )\n$$\nOur goal is to design distributed algorithms in order to minimize the regret.\n\\subsection{Related Work}\nSingle-player multi-armed bandits were originally proposed by \\cite{thompson} and asymptotically optimal strategies proposed by \\cite{lairobbins}, \\cite{klucb} amongst others. Building on the work of \\cite{lairobbins}, \\cite{anantharam} considered a generalization to the case where a single player can choose several arms, and proposed asymptotically optimal strategies. In fact, the problem considered by \\cite{anantharam} can be seen as a centralized version of the distributed problem we consider here. Motivated by wireless communication and networking, more recent works have considered multi-player multi-armed bandits, \\cite{liuzhao}, \\cite{rosenski}, for instance as a natural model for a scenario where several players must access a wireless channel in a decentralized manner \\cite{besson_multiplayer_revisited}. When collision sensing is available, several authors \\cite{boursier_sicmmab}, \\cite{wang} have shown that one can obtain roughly the same performance as in the centralized case by cleverly exploiting collisions. A harder problem is the case without collision sensing. The state-of-the-art known algorithms for this problem are SIC-MMAB2 \\cite{boursier_sicmmab}, EC-SIC \\cite{shi_ec_sic} and the algorithms of \\cite{lugosi}. Our work improves on these algorithms, both in terms of performance, and in terms of practical applicability. Some generalizations and alternative settings were also considered. \\cite{bistritz} and \\cite{mehrabian} consider the more general heterogenous case where the mean of arms vary among players. \\cite{magesh} and \\cite{shi_ec3} consider a variant of the probem where collisions can yield a non-zero reward. \\cite{bubeck2} consider the additional assumption that players have access to shared randomness, and study the minimax regret.\n\n\\subsection{Our Contribution}\n\nThe regret upper bounds for state-of-the-art algorithms and the prior information they require is presented in Table~\\ref{table:stateoftheart} (\\footnote{where $E$ in the EC-SIC regret bound is the error exponenent of the error correcting code used.}). We also present a regret lower bound which corresponds to the case with collision sensing, although we do not know if this bound is attainable without collision sensing. These algorithms suffer from two important issues: (i) They require a lower bound on the expected reward of the worst arm $\\mu_{(K)}$, and cannot be used without this prior information. Since in practice this information is usually not available, this limits their practical applicability. In fact EC-SIC also requires as prior information the gap $\\mu_{(M)} - \\mu_{(M+1)}$ which is also typically unknown (ii) Their regret is proportional to $O(1\/\\mu_{(K)})$ which can be arbitrarily large. Namely $\\mu_{(K)}$ may be very small, for instance, in wireless applications where arms represent channels, where $\\mu_{k}$ is an increasing function of the signal-to-noise ratio on channel $k$, and where $\\mu_1,...,\\mu_K$ are weakly correleated due to frequency selective fading, this issue occurs. Indeed, if we assume that $\\mu_1,...,\\mu_K$ are drawn independently from some distribution over $[0,1]$, then $\\mu_{(K)} = \\min_{k=1,...,K} \\mu_k$ is close to $0$ with overwhelming probability when $K$ is large.\n\nOur main contribution is to propose an algorithm correcting issues (i) and (ii). The proposed algorithm requires no input information besides the number of arms $K$ and the time horizon $T$ and its regret is upper bounded as:\n\\begin{align*}\n R(T,\\mu) = O\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + K^2 M \\ln T \n + K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln T \\Big)\n\\end{align*}\nwhich does not depend on $\\mu_{(K)}$. We also present numerical experiments which confirm our theoretical predictions. We believe that our work is a step towards closing the gap between the best known upper bound and lower bound, by showing that, just like in the case with collision sensing, the regret need not depend on $\\mu_{(K)}$, and that is it not necessary to have prior information about $\\mu_{(K)}$ either. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline \nAlgorithm & Regret Upper Bound \\\\ \n\\hline \nSIC-MMAB2 & $O\\Big(M \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + {K^2 M \\over \\mu_{(K)}} \\ln T \\Big)$ \\\\ \nEC-SIC & $O\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + {K M \\over \\mu_{(K)}} \\ln T + {K M^2 \\over E(\\mu_{(K)})} \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right) \\ln T \\Big)$ \\\\ \nLugosi et al. & $O\\Big( {K M \\over (\\mu_{(M)} - \\mu_{(M+1)})^2} \\ln T \\Big)$ (first algorithm) \\\\\nLugosi et al. & $O\\Big( {K M \\over \\mu_{(M)} - \\mu_{(M+1)}} \\ln T + {K^2 M \\over \\mu_{(M)}} (\\ln T)^2 \\Big)$ (second algorithm) \\\\\nOur work & $O\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + K^2 M \\ln T + K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln T \\Big)$ \\\\ \nLower bound & $\\Omega\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} \\Big)$ \\\\ \n\\hline \n\\end{tabular} \n\\vspace{1cm}\n\\begin{tabular}{|c|c|}\n\\hline \nAlgorithm & Prior Information \\\\ \n\\hline \nSIC-MMAB2 & $\\mu_{(K)}$ \\\\ \nEC-SIC & $\\mu_{(K)}$ and $\\mu_{(M)} - \\mu_{(M+1)}$ \\\\ \nLugosi et al. & $\\mu_{(M)}$ and $M$ \\\\ \nOur work & $\\emptyset$ \\\\ \n\\hline \n\\end{tabular} \n\\end{center}\n\\caption{State-of-the-art without collision sensing: regret upper bounds and prior information}\n\\label{table:stateoftheart}\n\\end{table}\n\n\\section{Algorithm and Analysis}\n\nThe proposed algorithm can be broken down into three steps and corresponding subroutines. Of course, since the proposed algorithm is distributed, all players $m=1,...,M$ apply the same algorithm based on their own observations. When providing pseudo-code, it is understood that each player follows the same pseudo code, in a distributed fashion. When stating our algorithms we use the notation $X \\mathrel{+}= x$ to denote $X \\gets X + x$, i.e. add $x$ to $X$. All additional subroutines as well as proofs are presented in the appendix.\n\n\\subsection{Step 1: Agreement on a Good Arm}\n\nThe first step of the proposed algorithm is the \\alg{FindGoodArm} subroutine. The goal of this subroutine is for all players $m=1,...,M$ to agree on a \"good\" arm $\\tilde{k}$, in the sense that $\\mu_{\\tilde{k}} \\ge (1\/8) \\mu_{(1)}$, as well as to output a lower bound on $\\mu_{\\tilde{k}}$.\n\nThis subroutine is the cornerstone of our algorithm, and the main reason why our algorithm does not suffer from (i) a regret scaling inversely proportionally to the reward of the first arm $O(1 \/ \\mu_{(K)})$ (ii) requiring a non trivial lower bound of $\\mu_{(K)}$ as an input parameter, while all state-of-the-art algorithms suffer from problems (i) and (ii). Indeed, once a good arm is identified, one can use this arm to exchange information between players, and arms with very low rewards have no bearing on performance and can be mostly ignored.\n\nThe \\alg{FindGoodArm} subroutine is designed such that, with high probability, not only will all players agree on the same good arm, but also that they will all terminate the procedure at the same time. This is important to maintain synchronization, and achieving this synchronization is not trivial as shown next.\n\nThe \\alg{FindGoodArm} subroutine is broken in successive phases, and each phase $p \\ge 1$ has two sub-phases. In the first sub-phase of phase $p$, each player selects an arm uniformly at random \n$6 K 2^{p} \\ln {2 \\over \\delta}$ times and observes the resulting rewards. For each arm $k=1,...,K$, each player computes \n$\n\\hat{\\mu}_k^p\n$\nwhich is the ratio between the sum of rewards obtained from arm $k$ and the number of times that arm $k$ has been selected. If $\\hat{\\mu}_k^p \\ge 2^{1-p}$, we say that the player accepts arm $k$ at phase $p$ (i.e. he believes that $\\mu_k$ is greater than $2^{-p}$) and otherwise he rejects arm $k$ at phase $p$. In the second sub-phase, the players attempt to reach agreement on one good arm. For all arms $\\ell=1,...,K$, if a player accepts arm $\\ell$ at phase $p$, he chooses an arm uniformly at random $K 2^{p} \\ln {2 \\over \\delta}$ times and if he does not accept $\\ell$ then he chooses arm $\\ell$ for $K 2^{p} \\ln {2 \\over \\delta}$ times. If arm $\\ell$ yields at least one non-zero reward then we say that the player confirms arm $\\ell$ in phase $p$. If there exists $\\tilde{k}$ such that the player confirms $\\tilde{k}$, then the procedure stops and $\\tilde{k}$ is output. This mechanism is designed so that, with high probability, the procedure does not exit until all players have agreed on a good arm. Indeed, if at least one of the players rejects arm $\\ell$, then he will select arm $\\ell$ all the time, and no non-zero reward can be obtained from $\\ell$, preventing other players from confirming arm $\\ell$.\n\nTo summarize the \\alg{FindGoodArm} subroutine involves sampling arms uniformly at random more and more, until at least one of them has a large enough reward, and periodically exchanging information between players in order to reach agreement over one good arm, and once agreement is reached, the procedure stops. Lemma~\\ref{lem:findgoodarm} provides an analysis of \\alg{FindGoodArm} and Algorithm~\\ref{algo:FindGoodArm} provides its pseudo-code.\n\n\\begin{lemma}\\label{lem:findgoodarm}\n Consider the \\alg{FindGoodArm} subroutine with input parameters $K$ and $\\delta$. Then, with probability greater than $1 - C_1 K M (\\ln {1 \\over \\mu_{(1)}}) \\delta$: (i) The subroutine lasts at most $C_2 {K^2 \\over \\mu_{(1)}} \\ln {1 \\over \\delta}$ time slots causing at most $C_2 K^2 M \\ln {1 \\over \\delta}$ regret. (ii) The output $\\tilde{k}$ is a \"good\" arm i.e. $\\mu_{\\tilde{k}} \\ge {1 \\over 8} \\mu_{(1)}$ (iii) The output $\\tilde{\\mu}$ is a lower bound on the expected reward of $\\tilde{k}$ i.e. $\\mu_{\\tilde{k}} \\ge \\tilde{\\mu}$ (vi) All players exit the subroutine at exactly the same time and output the same arm $\\tilde{k}$ and estimate $\\tilde{\\mu}$, where $C_1,C_2$ are universal constants.\n\\end{lemma}\n\n\\begin{algorithm}[h]\n\\caption{\\alg{FindGoodArm} (for player $m=1,...,M)$)}\n\\label{algo:FindGoodArm}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms , $\\delta$: confidence parameter\n\\ENSURE $\\tilde{k}$: good arm, $\\tilde{\\mu}$: lower bound on reward of $\\tilde{k}$\n\\STATE $p \\gets 0$, $\\tilde{k} \\gets -1$ {\\it \\# initialization} \n\\WHILE{$\\tilde{k} = -1$} \n \\STATE $p \\mathrel{+}= 1$, $R[k],N[k] \\gets 0$ for $k=1,...,K$ {\\it \\# current phase, rewards and number of samples}\n \\STATE{\\it \\# sub-phase 1: explore arms uniformly at random}\n \\FOR{$t = 1,...,6 K 2^{p} \\ln {2 \\over \\delta}$} \n \\STATE Select arm $k \\in \\{1,\\dots,K\\}$ uniformly at random, observe reward $r$, $R[k] \\mathrel{+}= r$, $N[k] \\mathrel{+}= 1$\n \\ENDFOR\n \\STATE{\\it \\# sub-phase 2: confirm accepted arms}\n \\FOR{$\\ell \\gets 1,\\dots,K$}\n \\STATE $R'[k] \\gets 0$ for $k=1,...,K$ {\\it \\# rewards of samples} \n \\STATE{\\it \\# if arm $\\ell$ was accepted sample arms uniformly}\n \\IF{${R[\\ell] \\over N[\\ell]} \\ge 2^{1-p}$}\n \\FOR{$t = 1,\\dots, 2^{p} K \\ln {2 \\over \\delta}$}\n \\STATE Select arm $k \\in \\{1,\\dots,K\\}$ uniformly at random and observe reward $r$, $R'[k] \\mathrel{+}= r$\n \\ENDFOR\n \\STATE{\\it \\# if a non-zero reward is obtained, confirm arm $\\ell$}\n \\ifthen{$R'[\\ell] \\ge 1$}{$\\tilde{k} \\gets \\ell$, $\\tilde{\\mu} \\gets 2^{-p}$ {\\bf break}} \n \\STATE{\\it \\hspace{-0.4cm} \\# if arm $\\ell$ was rejected only sample $\\ell$}\n \\ELSE\n \\for{$t =1,\\dots,2^{p} K \\ln {2 \\over \\delta}$}{select arm $k=\\ell$, observe reward $r$, $R'[k] \\mathrel{+}= r$} \n \\ENDIF\n \\ENDFOR\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Step 2: Rank Assignment}\nThe second part of our algorithm is the subroutine \\alg{ VirtualMusicalChairs} which performs rank assignment, that is, assigning a different number to each of the players. This allows to perform orthogonalization and explore arms without collisions by using sequential hopping as done in~\\cite{boursier_sicmmab}.\n\nOur subroutine \\alg{VirtualMusicalChairs} is based on the so-called Musical Chairs algorithm where players repeatedly choose an arm at random until they obtain a non-zero reward and then keep sampling this very same arm, which will also be their external rank. However, the Musical Chairs used in \\cite{boursier_sicmmab} is not satisfactory since it samples from all arms $k=1,...,K$. Its performance is therefore limited by the worst arm leading to a regret which scales proportionally to $O(1 \/ \\mu_{(K)})$, in addition to requiring a lower bound on $\\mu_{(K)}$ as an input parameter.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\linewidth]{figure_virtual_arms.pdf}\n \\caption{Mapping between a single real arm (below) and $K$ identical virtual arms (above).}\n \\label{figurevirtual}\n\\end{figure}\n\nInstead we propose to adapt this algorithm to only sample from the unique good arm $\\tilde{k}$ derived by the \\alg{FindGoodArm} subroutine. To do so we map this unique good arm into $K$ \"virtual\" arms by using a time division technique depicted in Figure~\\ref{figurevirtual}. We divide time in $\\tau$ blocks of length $K$. The $\\ell$-th time slot of each block is mapped to the $\\ell$-th virtual arm. At the beginning of a block, player $j$ chooses a virtual arm $\\ell$ at random (without pulling it). At the $\\ell$-th time slot of the block, he pulls the good arm $\\tilde{k}$, and if the reward is positive, he settles on virtual arm $\\ell$ until the end. Note that a player actually selects one arm every $K$ time steps on average. Note also that, due to this idea of virtual arms, \\alg{VirtualMusicalChairs} can be analyzed simply by considering the classical musical chairs algorithm applied to $K$ identical arms with mean reward $\\mu_{\\tilde{k}}$. Lemma~\\ref{lem:virtualmusicalchairs} provides an analysis of \\alg{VirtualMusicalChairs} and Algorithm~\\ref{algo:VirtualMusicalChairs} provides its pseudo-code.\n\n\\begin{lemma}\\label{lem:virtualmusicalchairs}\n Consider $\\delta > 0$. Consider the subroutine \\alg{VirtualMusicalChairs} with input parameters $K$, a good arm $\\tilde{k}$ and $\\tau = {K \\over \\mu_{\\tilde{k}}} \\ln {1 \\over \\delta}$. Then the subroutine exits after $K \\tau$ time slots, incurring regret at most $8 K^2 M \\ln {1 \\over \\delta}$ and with probability greater than $1 - C_3 K M \\delta$ it assigns a distinct external rank to all of the $M$ players where $C_3$ is a universal constant. \n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{\\alg{VirtualMusicalChairs} (for player $m=1,...,M$)}\n\\label{algo:VirtualMusicalChairs}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $\\tilde{k}$: a good arm, $\\tau$: sampling times\n\\ENSURE $s$: external rank of the player\n\\STATE $s \\gets -1$; { \\it \\# rank of the player is initially unset}\n\\STATE{ \\it \\# musical chairs on the arm $\\tilde{k}$}\n\\FOR{$t \\gets 1,\\dots,K \\tau$}\n \\STATE{ \\it \\# time is split in blocks of size $K$ and we select when to sample at the start of a block.}\n \\IF{$\\mod{(t,K)} = 1$}\n \\IF{$s = -1$}\n \\STATE Draw $\\ell \\in \\{1,...,K\\}$ uniformly at random { \\it \\# Choose a random slot if rank is unset}\n \\ELSE\n \\STATE $\\ell \\gets s$ { \\it \\# Choose the rank as a slot if it is set}\n \\ENDIF\n \\ENDIF\n \\STATE{ \\it \\# sample the corresponding time slot}\n \\IF{$\\mod{(t,K)} = \\ell$}\n \\STATE Select arm $\\tilde{k}$, and observe reward $r$\n \\STATE{ \\it \\# set rank if it was not set yet and a non zero reward was obtained}\n \\ifthen{$r > 0$ and $s=-1$}{$s \\gets \\ell$}\n \\ELSE \n \\STATE Select an arbitrary arm in $\\{1,...,K\\} \\setminus \\{\\tilde{k}\\}$ \n \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Step 3: Computation of the Number of Players}\n\nThe third step of the proposed algorithm is the \\alg{VirtualNumberPlayers} subroutine which allows players to estimate the number of players $M$. \\alg{VirtualNumberPlayers} takes as input $K$ the number of arms, $\\tilde{k}$ a good arm, $s$ the external rank of a player, $\\tau$ a sampling time, all of which are available from the previous subroutines. \n\nOnce again we combine the sequential hopping technique used by \\cite{boursier_sicmmab} to estimate $M$ along with the idea of virtual arms, so that the procedure only samples from a good arm $\\tilde{k}$, and hence is not impacted by the presence of some arms with low reward.\n\nTo estimate $M$, we initialize our estimate as $\\hat{M}=1$, and perform sequential hopping for $2K$ rounds to make sure that every two different players only collide one time. Since players can be orthogonalized thanks to their external rank, a player increments $\\hat{M}$ only if he gets all zero rewards during a specific period (this suggests that some collisions have occurred). As a by-product, we also output the internal rank $j \\in \\{1,...,M\\}$, not to be confused with the external rank $s \\in \\{1,...,K\\}$. Each player is assigned a different internal rank, and it will serve for the last step of our algorithm, in order to assign roles when exploring the various arms.\n\n\\begin{lemma}\\label{lem:virtualnumberplayers}\n Consider $\\delta > 0$. Consider the subroutine \\alg{VirtualNumberPlayers} with input parameters $K$, a good arm $\\tilde{k}$ and $\\tau = {1 \\over \\mu_{\\tilde{k}}} \\ln {1 \\over \\delta}$. Then the subroutine exits after $K^2 \\tau$ time slots causing at most $C_4 K^2 M \\ln {1 \\over \\delta}$ regret, \n and with probability greater than $1 - C_5 \\delta$, it both outputs a correct estimate of the number of players $\\hat{M} = M$, and it assigns a distinct internal rank $j \\in \\{1,...,M\\}$ to all of the $M$ players. Both $C_4$ and $C_5$ are universal constants. \n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{\\alg{VirtualNumberPlayers} (for player $m=1,...,M$)}\n\\label{alg:NumberPlayers}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $\\tilde{k}$ a good arm, $s$: external rank of a player, $\\tau$: sampling times\n\\ENSURE $\\hat{M}$: estimated number of players, $j$: internal rank of the player\n\\STATE $\\hat{M} \\gets 1$, $\\ell \\gets s$, $j \\gets 1$ {\\it \\# initialization}\n\\FOR{$n=1,\\dots,2K$}\n \\ifthen{$n > 2s$}{$\\ell \\gets \\mod{(\\ell+1,K)}$ {\\it \\# sequential hopping}}\n \\STATE $R \\gets 0$ {\\it \\# sum of rewards from the good arm}\n \\STATE {\\it \\# sample from the good arm}\n \\FOR{$k = 1,\\dots,K$}\n \\IF{$\\ell \\ne k$}\n \\for{$t=1,\\dots,\\tau$}{Select an arbitrary arm in $\\{1,...,K\\} \\setminus \\{\\tilde{k}\\}$}\n \\ELSE\n \\STATE {\\it \\# sample from virtual arm $\\tau$ times}\n \\for{$t=1,\\dots,\\tau$}{Select arm $\\tilde{k}$, observe reward $r$, $R \\mathrel{+}= r$} \\STATE {\\it \\# if no non-zero reward was obtained increase the estimated number of players}\n \\IF{$R = 0$}\n \\STATE $\\hat{M} \\mathrel{+}= 1$, \n \\ifthen{$n \\le 2s$}{$j \\mathrel{+}= 1$}\n \\ENDIF\n \\ENDIF\n \\ENDFOR\n\\ENDFOR\n\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Step 4: Distributed Exploration and Finding the Best Arms}\n\nThe last step is the subroutine \\alg{DistributedExploration}, which takes as input $K$ the number of arms, $j$ the internal rank of the player, $\\tau$ the sampling times, $\\hat{M}$ the estimated number of players, and outputs one arm amongst the $M$ best arms, which is assigned to the player. \n\n\\alg{DistributedExploration} is similar to the exploration strategy used in \\cite{shi_ec_sic} \nwith two key improvements. First, to avoid choosing arms with low expected reward, a player sends data only through arm $\\tilde{k}$ rather than through the $j$-th arm as done in \\cite{shi_ec_sic}. Second, we set the quantization error to be dependent on phase p instead of the fixed quantity $\\mu_{M} - \\mu_{M+1}$. By doing so, the number of pulls is reduced, and the subroutine does not require this prior knowledge anymore.\nIts analysis, provided in lemma~\\ref{lem:distributedexploration} follows along similar lines as \\cite{shi_ec_sic}, so we simply highlight the main steps of this procedure. The player with internal rank $j=1$ is called the leader, and other players are called the followers. The procedure operates in phases, and in phase $p$, the active players (the players that have not been assigned an arm yet) sample each active arm (the arms for which players have not figured out whether or not they are amongst the $M$ best arms) $2^p \\lceil \\ln {1 \\over \\delta} \\rceil$ times using sequential hopping (so that no collisions occur). Then all followers send their estimates to the leader, the leader aggregates their estimates, determines which arms can be accepted (the arms for which one is sure that they are amongst the $M$ best arms) and rejected (the arms for which one is sure that they are not amongst the $M$ best arms) and then sends the sets of accepted and rejected arms back to the followers. If an arm is accepted then it is assigned to an active player, and both the arm and the player become inactive, and the player simply selects the assigned arm until the end of the procedure. If an arm is rejected then it becomes inactive. The strategy on how leader and followers act in this information exchange step is given in subroutines \\alg{ComFollow} and \\alg{ComLead} in appendix. In essence, the leader coordinates and takes all of the decisions, while followers simply collect samples, transmit their estimates and receive orders on what to do next. The strategy to send and receive data between leader and followers is given in subroutines \\alg{EncoderSendFloat} and \\alg{DecoderReceiveFloat} in appendix. \n\\begin{lemma}\\label{lem:distributedexploration}\n Consider $\\delta > 0$ and $T \\ge 0$. Consider the subroutine \\alg{DistributedExploration} with input parameters $K$, $\\tilde{k}$ and $\\tau = {1 \\over \\mu_{\\tilde{k}}} \\ln {1 \\over \\delta}$ applied with time horizon $T$ (\\footnote{Applying a procedure with time horizon $T$ simply means that, if it does not terminate before $T$ on its own, then its execution is immediately stopped when time $T$ is reached.}). Then with probability greater that $1 - C_6 \\delta K M (\\ln T)^2 $ the subroutine incurs a regret upper bounded by\n \\begin{align*}\n C_7 & \\sum_{k > M} {\\ln {1 \\over \\delta} \\over \\mu_{(M)} - \\mu_{(k)}} + C_{8} K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln {1 \\over \\delta}\n \\end{align*}\n and assigns one distinct arm to each player from the set of the $M$ best arms, where $C_6,C_7,C_8$ are universal constants.\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{\\alg{DistributedExploration} (for players $m=1,...,M$)}\n\\label{algo}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $j$: internal rank of a player, $\\hat{M}$: number of players, $\\tilde{k}$: a good arm, $\\tau$: sampling times\n\\ENSURE $f$ an arm amongst the $M$ best arms assigned to the player\n\\STATE Initialize $p \\gets 0$; $f \\gets -1$; {\\it \\# initialization}\n\\STATE $R[k], v[k] \\gets 0$ for $k=1,...,K$ {\\it \\# rewards and number of samples for each arm}\n\\STATE {\\it \\# rewards and number of samples for each arm held by each players, only stored by the leader}\n\\IF{$j=1$}\n \\for{$m=1,...,\\hat{M}$ and $k=1,...,K$}{$\\hat\\mu[k,m],N[k,m] \\gets 0$}\n\\ENDIF\n\\STATE $M' \\gets \\hat{M}$, ${\\cal K} \\gets \\{1,\\dots,K\\}$ {\\it \\# number of active players and set of active arms}\n\\WHILE{$f = -1$}\n\\STATE $p \\mathrel{+}= 1$ {\\it \\# start phase $p$}\n\\STATE $k \\gets j$ {\\it \\# first sub-phase explore arms by sequential hopping}\n\\FOR{$t \\gets 1,\\dots, |\\mathcal K| 2^{p} \\left\\lceil \\ln {1 \\over \\delta} \\right\\rceil$}\n\\STATE $k \\gets (k + 1) \\mod |\\mathcal K|$ \n\\STATE Select arm $k$, observe reward $r$, $R[k] \\mathrel{+}= r$, $v[k]\\mathrel{+}= 1$, $E[k] \\gets {R[k] \\over v[k]}$ \n\\ENDFOR\n\\STATE $Q \\gets \\lceil {p \\over 2} + 3 \\rceil$ {\\it \\# second sub-phase: share estimates between players}\n\\IF{$j = 1$}\n\\STATE $(f,{\\cal K},M',\\hat{\\mu},N) \\gets $\\alg{ComLeader}$(\\hat{\\mu},N,{\\cal K},M',Q,\\tau,$ $ \\tilde{k},p,\\delta)$ {\\# player is a leader}\n\\ELSE \n\\STATE $(f,\\mathcal {\\cal K},M') \\gets $\\alg{ComFollow}$(E,j,\\mathcal {\\cal K},M',Q,\\tau,\\tilde{k})$ {\\# player is a follower}\n\\ENDIF\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Putting it all Together}\n\nThe complete proposed algorithm is presented in algorithm~\\ref{alg:proposed}, and combines the four steps above. It is noted that our proposed algorithm is in fact a procedure to identify the set of best arms. To turn it into an algorithm for minimizing regret over a time horizon of $T$, one can simply run the proposed algorithm with confidence parameter $\\delta = {1 \\over T}$, and then let players select the arm that has been assigned to them by the algorithm until the time horizon runs out. The regret upper bound of Theorem~\\ref{th:main} is our main result, and is simply proven by combining the four previous lemmas. As promised, unlike state-of-the-art algorithms, our algorithm does not need any prior information such as the number of arms $K$ or the reward of the worst arm $\\mu_{(K)}$, and its regret does not depend on $\\mu_{(K)}$. The regret bound is better than that of SIC-MMAB2 and EC-SIC, in the sense that the term proportional to $1 \/ \\mu_{(K)}$ (which can be arbitrarily large), has been eliminated. In other words, performance is not limited by the worst arm anymore. This causes a dramatic performance gain which is seen in numerical experiments shown below.\n\n\\begin{theorem}\\label{th:main}\n Consider $T \\ge 0$, $\\mu_{M}-\\mu_{M+1}>0$. First apply the proposed algorithm with input parameters $K$ and $\\delta = {1 \\over T (\\ln T)}$ with time horizon $T$, let $\\bar{k}$ denote its output, and select arm $\\bar{k}$ for the remaining time steps. Then the expected regret of this procedure is \n \\begin{align*}\n R(T) &\\le C_{9} \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + C_{10} K^2 M \\ln T + C_{11} K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln T\n \\end{align*}\n with $C_{9},C_{10},C_{11}$ three universal constants.\n\\end{theorem}\n\n\n\\begin{algorithm}\n\\caption{Proposed algorithm (for player $m=1,...,M$)}\n\\label{alg:proposed}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $\\delta$: confidence level \n\\ENSURE $\\bar{k}$ an arm amongst the $M$ best arms assigned to the player\n\\STATE $(\\tilde{k}, \\tilde{\\mu}) \\gets $\\alg{FindGoodArm}$(K,\\delta)$ {\\it \\# find a good arm and a lower bound on its reward}\n\\STATE $s \\gets $\\alg{VirtualMusicalChairs}$(K, \\tilde{k}, K \\ln(\\frac{1}{\\delta})\/\\tilde{\\mu})$ {\\it \\# assign external rank to each player}\n\\STATE $(\\hat{M},j) \\gets $\\alg{VirtualNumberPlayers}$(K, \\tilde{k}, s, \\ln(\\frac{1}{\\delta}) \/ \\tilde{\\mu})$ {\\it \\# estimate the number of players and assign internal rank}\n\\STATE $\\bar{k} \\gets $\\alg{DistributedExploration}$(K,j,\\hat{M},\\tilde{k}, \\ln(\\frac{1}{\\delta}) \/ \\tilde{\\mu}))$ {\\it \\# find one arms out of the $M$ best arms}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Numerical Experiments}\n\n\nWe now compare the empirical performance of the proposed algorithm against the state-of-the-art algorithms SIC-MMAB2 and EC-SIC. For simplicity we assume that rewards decrease from best to worst in a linear fashion $\\mu_{(k)} = \\mu_{(1)} + {k - 1 \\over K-1} (\\mu_{(K)} - \\mu_{(1)})$. The regret of algorithms is averaged over $20$ (or more) independent runs, and $95\\%$ confidence intervals are presented. \n\n{\\bf Influence of the number of players} In our first set of experiments, we consider $\\mu_{(1)} = 1$ and $\\mu_{(K)} = 0.01$, $M = \\lfloor K\/2 \\rfloor$. We plot the expected regret of the various algorithms for $K = 5,10,20$ in figures \\ref{figureK5M2},\\ref{figureK10M5} and \\ref{figureK20M10} respectively. Overall, the proposed algorithm clearly outperforms EC-SIC and SIC-MMAB2, sometimes by several orders of magnitude, and the difference seems more and more severe when $K$ increases.\n\n{\\bf Influence of the gap} Figure \\ref{figureK5M2} and \\ref{figureK10M5} consider $\\mu_{M}-\\mu_{M+1} = 0.2$ and $\\mu_{M}-\\mu_{M+1} = 0.1$. We see that sometimes the proposed algorithm performs a bit worse than other algorithms for very small time horizons, however for larger time horizons it does outperform SIC-MMAB2 and EC-SIC which seem not to converge quickly enough.\n\n\\begin{figure}[H]\n \\centering\n \\begin{minipage}[t]{0.3\\linewidth}\n \\includegraphics[width=1\\linewidth]{K5M2.pdf}\n \\caption{K=5,M=2,T=$10^5$}\n \\label{figureK5M2}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K10M5.pdf}\n \\caption{K=10,M=5,T=$10^6$}\n \\label{figureK10M5}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K20M10.pdf}\n \\caption{K=20,M=10,T=$10^7$}\n \\label{figureK20M10}\n \\end{minipage}\n\\end{figure}\n\n{\\bf Impact of the number of players} In figures \\ref{figureK10M2} and \\ref{figureK10M8} we compare between $M=2$ players and $M=8$ players. For $M=2$, both the proposed algorithm and EC-SIC do converge. However, when $M$ increases, the collision probability becomes larger, causing EC-SIC to spend significant time performing musical chairs to estimate $M$. \n\\begin{figure}[H]\n \\centering\n \\begin{minipage}[t]{0.3\\linewidth}\n \\includegraphics[width=1\\linewidth]{K10M2.pdf}\n \\caption{K=10,M=2,T=$10^6$}\n \\label{figureK10M2}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K10M8.pdf}\n \\caption{K=10,M=8,T=$10^6$}\n \\label{figureK10M8}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K5M2vsmu.pdf}\n \\caption{K=5,M=2,T=$10^5$ with different $\\mu_{(K)}$}\n \\label{figureK5M2mu}\n \\end{minipage}\n\\end{figure}\n\n\n{\\bf Impact of the worst arm} We present the regret for different values of $\\mu_{(K)}$ in figure \\ref{figureK5M2mu}. When $\\mu_{(K)}$ decreases, we can see that the performance of SIC-MMAB2 and EC-SIC is greatly affected, and for $\\mu_{(K)} = 0.001$ they do not even start the exploration phase. On the other hand, no matter how small $\\mu_{(K)}$ is, the cumulative regret of the proposed algorithm stays similar, which confirms our theoretical predictions.\n\n\\section{Conclusion}\n\nIn this work we have proposed a new algorithm for multi-player multi-armed bandits without collision sensing information. Through both analysis and numerical experiments, we have proven that it performs significantly better than the state-of-the-art algorithms, while at the same time requiring no input parameter other than the number of arms. The cornerstone of our approach is a novel, non trivial distributed procedure to enable players to discover a good arm without any prior information other than the number of arms. \nWe believe that this result is a significant contribution to the problem as it constitutes a step towards being able to solve multi-player multi-armed bandits in a practical setting (for instance cognitive radios) where no prior information is known about the expected rewards of the arms.\n\n{\\bf Acknowledgement:} The work of Wei Huang was supported by Beyond 5G, a project of the French Government's recovery plan ``France Relance''.\n\n\\newpage \n\n\\section{Introduction and Contribution}\n\n\\subsection{Multi-player Bandits}\nWe consider the multi-player multi-armed bandit problem with $K$ arms and $M < K$ players. At each time $t=1,...,T$, where $T$ is the time horizon, each player $m=1,...,M$ selects an arm denoted by $\\pi_m(t) \\in \\{1,...,K\\}$. If arm $k=1,...,K$ has been selected by strictly more than one player we say that a collision occurs on arm $k$, and we denote by \n$\n\\eta_{k}(t) = \\mathbbm{1} \\left\\{ \\sum_{m=1}^{M} \\mathbbm{1}\\{ \\pi_m(t) = k \\} \\ge 2 \\right\\}\n$\nthe corresponding indicator function. The reward received by player $m=1,...,M$ is denoted by\n$\n r_m(t) = X_{\\pi_m(t)}(t) [1-\\eta_{\\pi_m(t)}(t)]\n$\nwhere $(X_k(t))_{t \\ge 1}$ is i.i.d. Bernoulli with mean $\\mu_k$ for $k=1,...,K$. In other words, if a player $m$ is involved in a collision then he receives a reward of $0$, otherwise he receives a binary reward with mean $\\mu_{\\pi_m(t)}$.\n\n\\subsection{No Collision Information}\n\nWe consider the case where each player $m=1,...,M$ only observes its own reward $r_m(t)$ and nothing else. He does not observe the rewards obtained by other players, he does not know which arms have been selected by other players, and he cannot observe the collision indicator $\\eta_{\\pi_m(t)}$. This case is called \"without collision sensing\", and is the most arduous because when player $m$ gets a reward of $r_m(t) = 0$, he does not know whether this was caused by a collision so that $\\eta_{\\pi_m(t)}(t) = 1$ or whether this was caused because the reward of the chosen arm was null i.e. $X_{\\pi_m(t)}(t) = 0$. Of course, when a reward of $r_m(t) = 1$ is obtained, he can however be sure that no collision has occurred. Therefore, the problem is fully decentralized, and no communication is allowed between the players.\n\n\\subsection{Regret Minimization}\n\n Recall that $\\mu_1,...,\\mu_K$ denotes the mean reward of each arm where no collision has occurred, and define $\\mu_{(1)},....,\\mu_{(K)}$ its version sorted in descending order, so that $\\mu_{(1)} \\ge \\mu_{(2)} \\ge ... \\ge \\mu_{(K)}.$ The goal for the players is to maximize the expected sum of rewards, and to do so, each player must select a distinct arm amongst the $M$ best arms to both avoid any collisions and maximize total reward. The regret (sometimes called pseudo-regret) is defined as the difference between the total expected reward obtained by an oracle that plays the optimal decision and the reward obtained by the players\n$$\n R(T,\\mu) = \\sum_{t=1}^{T} \\sum_{m=1}^{M} \\mu_{(m)} - \\mathbb{E}( r_{m}(t) )\n$$\nOur goal is to design distributed algorithms in order to minimize the regret.\n\\subsection{Related Work}\nSingle-player multi-armed bandits were originally proposed by \\cite{thompson} and asymptotically optimal strategies proposed by \\cite{lairobbins}, \\cite{klucb} amongst others. Building on the work of \\cite{lairobbins}, \\cite{anantharam} considered a generalization to the case where a single player can choose several arms, and proposed asymptotically optimal strategies. In fact, the problem considered by \\cite{anantharam} can be seen as a centralized version of the distributed problem we consider here. Motivated by wireless communication and networking, more recent works have considered multi-player multi-armed bandits, \\cite{liuzhao}, \\cite{rosenski}, for instance as a natural model for a scenario where several players must access a wireless channel in a decentralized manner \\cite{besson_multiplayer_revisited}. When collision sensing is available, several authors \\cite{boursier_sicmmab}, \\cite{wang} have shown that one can obtain roughly the same performance as in the centralized case by cleverly exploiting collisions. A harder problem is the case without collision sensing. The state-of-the-art known algorithms for this problem are SIC-MMAB2 \\cite{boursier_sicmmab}, EC-SIC \\cite{shi_ec_sic} and the algorithms of \\cite{lugosi}. Our work improves on these algorithms, both in terms of performance, and in terms of practical applicability. Some generalizations and alternative settings were also considered. \\cite{bistritz} and \\cite{mehrabian} consider the more general heterogenous case where the mean of arms vary among players. \\cite{magesh} and \\cite{shi_ec3} consider a variant of the probem where collisions can yield a non-zero reward. \\cite{bubeck2} consider the additional assumption that players have access to shared randomness, and study the minimax regret.\n\n\\subsection{Our Contribution}\n\nThe regret upper bounds for state-of-the-art algorithms and the prior information they require is presented in Table~\\ref{table:stateoftheart} (\\footnote{where $E$ in the EC-SIC regret bound is the error exponenent of the error correcting code used.}). We also present a regret lower bound which corresponds to the case with collision sensing, although we do not know if this bound is attainable without collision sensing. These algorithms suffer from two important issues: (i) They require a lower bound on the expected reward of the worst arm $\\mu_{(K)}$, and cannot be used without this prior information. Since in practice this information is usually not available, this limits their practical applicability. In fact EC-SIC also requires as prior information the gap $\\mu_{(M)} - \\mu_{(M+1)}$ which is also typically unknown (ii) Their regret is proportional to $O(1\/\\mu_{(K)})$ which can be arbitrarily large. Namely $\\mu_{(K)}$ may be very small, for instance, in wireless applications where arms represent channels, where $\\mu_{k}$ is an increasing function of the signal-to-noise ratio on channel $k$, and where $\\mu_1,...,\\mu_K$ are weakly correleated due to frequency selective fading, this issue occurs. Indeed, if we assume that $\\mu_1,...,\\mu_K$ are drawn independently from some distribution over $[0,1]$, then $\\mu_{(K)} = \\min_{k=1,...,K} \\mu_k$ is close to $0$ with overwhelming probability when $K$ is large.\n\nOur main contribution is to propose an algorithm correcting issues (i) and (ii). The proposed algorithm requires no input information besides the number of arms $K$ and the time horizon $T$ and its regret is upper bounded as:\n\\begin{align*}\n R(T,\\mu) = O\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + K^2 M \\ln T \n + K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln T \\Big)\n\\end{align*}\nwhich does not depend on $\\mu_{(K)}$. We also present numerical experiments which confirm our theoretical predictions. We believe that our work is a step towards closing the gap between the best known upper bound and lower bound, by showing that, just like in the case with collision sensing, the regret need not depend on $\\mu_{(K)}$, and that is it not necessary to have prior information about $\\mu_{(K)}$ either. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline \nAlgorithm & Regret Upper Bound \\\\ \n\\hline \nSIC-MMAB2 & $O\\Big(M \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + {K^2 M \\over \\mu_{(K)}} \\ln T \\Big)$ \\\\ \nEC-SIC & $O\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + {K M \\over \\mu_{(K)}} \\ln T + {K M^2 \\over E(\\mu_{(K)})} \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right) \\ln T \\Big)$ \\\\ \nLugosi et al. & $O\\Big( {K M \\over (\\mu_{(M)} - \\mu_{(M+1)})^2} \\ln T \\Big)$ (first algorithm) \\\\\nLugosi et al. & $O\\Big( {K M \\over \\mu_{(M)} - \\mu_{(M+1)}} \\ln T + {K^2 M \\over \\mu_{(M)}} (\\ln T)^2 \\Big)$ (second algorithm) \\\\\nOur work & $O\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + K^2 M \\ln T + K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln T \\Big)$ \\\\ \nLower bound & $\\Omega\\Big( \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} \\Big)$ \\\\ \n\\hline \n\\end{tabular} \n\\vspace{1cm}\n\\begin{tabular}{|c|c|}\n\\hline \nAlgorithm & Prior Information \\\\ \n\\hline \nSIC-MMAB2 & $\\mu_{(K)}$ \\\\ \nEC-SIC & $\\mu_{(K)}$ and $\\mu_{(M)} - \\mu_{(M+1)}$ \\\\ \nLugosi et al. & $\\mu_{(M)}$ and $M$ \\\\ \nOur work & $\\emptyset$ \\\\ \n\\hline \n\\end{tabular} \n\\end{center}\n\\caption{State-of-the-art without collision sensing: regret upper bounds and prior information}\n\\label{table:stateoftheart}\n\\end{table}\n\n\\section{Algorithm and Analysis}\n\nThe proposed algorithm can be broken down into three steps and corresponding subroutines. Of course, since the proposed algorithm is distributed, all players $m=1,...,M$ apply the same algorithm based on their own observations. When providing pseudo-code, it is understood that each player follows the same pseudo code, in a distributed fashion. When stating our algorithms we use the notation $X \\mathrel{+}= x$ to denote $X \\gets X + x$, i.e. add $x$ to $X$. All additional subroutines as well as proofs are presented in the appendix.\n\n\\subsection{Step 1: Agreement on a Good Arm}\n\nThe first step of the proposed algorithm is the \\alg{FindGoodArm} subroutine. The goal of this subroutine is for all players $m=1,...,M$ to agree on a \"good\" arm $\\tilde{k}$, in the sense that $\\mu_{\\tilde{k}} \\ge (1\/8) \\mu_{(1)}$, as well as to output a lower bound on $\\mu_{\\tilde{k}}$.\n\nThis subroutine is the cornerstone of our algorithm, and the main reason why our algorithm does not suffer from (i) a regret scaling inversely proportionally to the reward of the first arm $O(1 \/ \\mu_{(K)})$ (ii) requiring a non trivial lower bound of $\\mu_{(K)}$ as an input parameter, while all state-of-the-art algorithms suffer from problems (i) and (ii). Indeed, once a good arm is identified, one can use this arm to exchange information between players, and arms with very low rewards have no bearing on performance and can be mostly ignored.\n\nThe \\alg{FindGoodArm} subroutine is designed such that, with high probability, not only will all players agree on the same good arm, but also that they will all terminate the procedure at the same time. This is important to maintain synchronization, and achieving this synchronization is not trivial as shown next.\n\nThe \\alg{FindGoodArm} subroutine is broken in successive phases, and each phase $p \\ge 1$ has two sub-phases. In the first sub-phase of phase $p$, each player selects an arm uniformly at random \n$6 K 2^{p} \\ln {2 \\over \\delta}$ times and observes the resulting rewards. For each arm $k=1,...,K$, each player computes \n$\n\\hat{\\mu}_k^p\n$\nwhich is the ratio between the sum of rewards obtained from arm $k$ and the number of times that arm $k$ has been selected. If $\\hat{\\mu}_k^p \\ge 2^{1-p}$, we say that the player accepts arm $k$ at phase $p$ (i.e. he believes that $\\mu_k$ is greater than $2^{-p}$) and otherwise he rejects arm $k$ at phase $p$. In the second sub-phase, the players attempt to reach agreement on one good arm. For all arms $\\ell=1,...,K$, if a player accepts arm $\\ell$ at phase $p$, he chooses an arm uniformly at random $K 2^{p} \\ln {2 \\over \\delta}$ times and if he does not accept $\\ell$ then he chooses arm $\\ell$ for $K 2^{p} \\ln {2 \\over \\delta}$ times. If arm $\\ell$ yields at least one non-zero reward then we say that the player confirms arm $\\ell$ in phase $p$. If there exists $\\tilde{k}$ such that the player confirms $\\tilde{k}$, then the procedure stops and $\\tilde{k}$ is output. This mechanism is designed so that, with high probability, the procedure does not exit until all players have agreed on a good arm. Indeed, if at least one of the players rejects arm $\\ell$, then he will select arm $\\ell$ all the time, and no non-zero reward can be obtained from $\\ell$, preventing other players from confirming arm $\\ell$.\n\nTo summarize the \\alg{FindGoodArm} subroutine involves sampling arms uniformly at random more and more, until at least one of them has a large enough reward, and periodically exchanging information between players in order to reach agreement over one good arm, and once agreement is reached, the procedure stops. Lemma~\\ref{lem:findgoodarm} provides an analysis of \\alg{FindGoodArm} and Algorithm~\\ref{algo:FindGoodArm} provides its pseudo-code.\n\n\\begin{lemma}\\label{lem:findgoodarm}\n Consider the \\alg{FindGoodArm} subroutine with input parameters $K$ and $\\delta$. Then, with probability greater than $1 - C_1 K M (\\ln {1 \\over \\mu_{(1)}}) \\delta$: (i) The subroutine lasts at most $C_2 {K^2 \\over \\mu_{(1)}} \\ln {1 \\over \\delta}$ time slots causing at most $C_2 K^2 M \\ln {1 \\over \\delta}$ regret. (ii) The output $\\tilde{k}$ is a \"good\" arm i.e. $\\mu_{\\tilde{k}} \\ge {1 \\over 8} \\mu_{(1)}$ (iii) The output $\\tilde{\\mu}$ is a lower bound on the expected reward of $\\tilde{k}$ i.e. $\\mu_{\\tilde{k}} \\ge \\tilde{\\mu}$ (vi) All players exit the subroutine at exactly the same time and output the same arm $\\tilde{k}$ and estimate $\\tilde{\\mu}$, where $C_1,C_2$ are universal constants.\n\\end{lemma}\n\n\\begin{algorithm}[h]\n\\caption{\\alg{FindGoodArm} (for player $m=1,...,M)$)}\n\\label{algo:FindGoodArm}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms , $\\delta$: confidence parameter\n\\ENSURE $\\tilde{k}$: good arm, $\\tilde{\\mu}$: lower bound on reward of $\\tilde{k}$\n\\STATE $p \\gets 0$, $\\tilde{k} \\gets -1$ {\\it \\# initialization} \n\\WHILE{$\\tilde{k} = -1$} \n \\STATE $p \\mathrel{+}= 1$, $R[k],N[k] \\gets 0$ for $k=1,...,K$ {\\it \\# current phase, rewards and number of samples}\n \\STATE{\\it \\# sub-phase 1: explore arms uniformly at random}\n \\FOR{$t = 1,...,6 K 2^{p} \\ln {2 \\over \\delta}$} \n \\STATE Select arm $k \\in \\{1,\\dots,K\\}$ uniformly at random, observe reward $r$, $R[k] \\mathrel{+}= r$, $N[k] \\mathrel{+}= 1$\n \\ENDFOR\n \\STATE{\\it \\# sub-phase 2: confirm accepted arms}\n \\FOR{$\\ell \\gets 1,\\dots,K$}\n \\STATE $R'[k] \\gets 0$ for $k=1,...,K$ {\\it \\# rewards of samples} \n \\STATE{\\it \\# if arm $\\ell$ was accepted sample arms uniformly}\n \\IF{${R[\\ell] \\over N[\\ell]} \\ge 2^{1-p}$}\n \\FOR{$t = 1,\\dots, 2^{p} K \\ln {2 \\over \\delta}$}\n \\STATE Select arm $k \\in \\{1,\\dots,K\\}$ uniformly at random and observe reward $r$, $R'[k] \\mathrel{+}= r$\n \\ENDFOR\n \\STATE{\\it \\# if a non-zero reward is obtained, confirm arm $\\ell$}\n \\ifthen{$R'[\\ell] \\ge 1$}{$\\tilde{k} \\gets \\ell$, $\\tilde{\\mu} \\gets 2^{-p}$ {\\bf break}} \n \\STATE{\\it \\hspace{-0.4cm} \\# if arm $\\ell$ was rejected only sample $\\ell$}\n \\ELSE\n \\for{$t =1,\\dots,2^{p} K \\ln {2 \\over \\delta}$}{select arm $k=\\ell$, observe reward $r$, $R'[k] \\mathrel{+}= r$} \n \\ENDIF\n \\ENDFOR\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Step 2: Rank Assignment}\nThe second part of our algorithm is the subroutine \\alg{ VirtualMusicalChairs} which performs rank assignment, that is, assigning a different number to each of the players. This allows to perform orthogonalization and explore arms without collisions by using sequential hopping as done in~\\cite{boursier_sicmmab}.\n\nOur subroutine \\alg{VirtualMusicalChairs} is based on the so-called Musical Chairs algorithm where players repeatedly choose an arm at random until they obtain a non-zero reward and then keep sampling this very same arm, which will also be their external rank. However, the Musical Chairs used in \\cite{boursier_sicmmab} is not satisfactory since it samples from all arms $k=1,...,K$. Its performance is therefore limited by the worst arm leading to a regret which scales proportionally to $O(1 \/ \\mu_{(K)})$, in addition to requiring a lower bound on $\\mu_{(K)}$ as an input parameter.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\linewidth]{figure_virtual_arms.pdf}\n \\caption{Mapping between a single real arm (below) and $K$ identical virtual arms (above).}\n \\label{figurevirtual}\n\\end{figure}\n\nInstead we propose to adapt this algorithm to only sample from the unique good arm $\\tilde{k}$ derived by the \\alg{FindGoodArm} subroutine. To do so we map this unique good arm into $K$ \"virtual\" arms by using a time division technique depicted in Figure~\\ref{figurevirtual}. We divide time in $\\tau$ blocks of length $K$. The $\\ell$-th time slot of each block is mapped to the $\\ell$-th virtual arm. At the beginning of a block, player $j$ chooses a virtual arm $\\ell$ at random (without pulling it). At the $\\ell$-th time slot of the block, he pulls the good arm $\\tilde{k}$, and if the reward is positive, he settles on virtual arm $\\ell$ until the end. Note that a player actually selects one arm every $K$ time steps on average. Note also that, due to this idea of virtual arms, \\alg{VirtualMusicalChairs} can be analyzed simply by considering the classical musical chairs algorithm applied to $K$ identical arms with mean reward $\\mu_{\\tilde{k}}$. Lemma~\\ref{lem:virtualmusicalchairs} provides an analysis of \\alg{VirtualMusicalChairs} and Algorithm~\\ref{algo:VirtualMusicalChairs} provides its pseudo-code.\n\n\\begin{lemma}\\label{lem:virtualmusicalchairs}\n Consider $\\delta > 0$. Consider the subroutine \\alg{VirtualMusicalChairs} with input parameters $K$, a good arm $\\tilde{k}$ and $\\tau = {K \\over \\mu_{\\tilde{k}}} \\ln {1 \\over \\delta}$. Then the subroutine exits after $K \\tau$ time slots, incurring regret at most $8 K^2 M \\ln {1 \\over \\delta}$ and with probability greater than $1 - C_3 K M \\delta$ it assigns a distinct external rank to all of the $M$ players where $C_3$ is a universal constant. \n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{\\alg{VirtualMusicalChairs} (for player $m=1,...,M$)}\n\\label{algo:VirtualMusicalChairs}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $\\tilde{k}$: a good arm, $\\tau$: sampling times\n\\ENSURE $s$: external rank of the player\n\\STATE $s \\gets -1$; { \\it \\# rank of the player is initially unset}\n\\STATE{ \\it \\# musical chairs on the arm $\\tilde{k}$}\n\\FOR{$t \\gets 1,\\dots,K \\tau$}\n \\STATE{ \\it \\# time is split in blocks of size $K$ and we select when to sample at the start of a block.}\n \\IF{$\\mod{(t,K)} = 1$}\n \\IF{$s = -1$}\n \\STATE Draw $\\ell \\in \\{1,...,K\\}$ uniformly at random { \\it \\# Choose a random slot if rank is unset}\n \\ELSE\n \\STATE $\\ell \\gets s$ { \\it \\# Choose the rank as a slot if it is set}\n \\ENDIF\n \\ENDIF\n \\STATE{ \\it \\# sample the corresponding time slot}\n \\IF{$\\mod{(t,K)} = \\ell$}\n \\STATE Select arm $\\tilde{k}$, and observe reward $r$\n \\STATE{ \\it \\# set rank if it was not set yet and a non zero reward was obtained}\n \\ifthen{$r > 0$ and $s=-1$}{$s \\gets \\ell$}\n \\ELSE \n \\STATE Select an arbitrary arm in $\\{1,...,K\\} \\setminus \\{\\tilde{k}\\}$ \n \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Step 3: Computation of the Number of Players}\n\nThe third step of the proposed algorithm is the \\alg{VirtualNumberPlayers} subroutine which allows players to estimate the number of players $M$. \\alg{VirtualNumberPlayers} takes as input $K$ the number of arms, $\\tilde{k}$ a good arm, $s$ the external rank of a player, $\\tau$ a sampling time, all of which are available from the previous subroutines. \n\nOnce again we combine the sequential hopping technique used by \\cite{boursier_sicmmab} to estimate $M$ along with the idea of virtual arms, so that the procedure only samples from a good arm $\\tilde{k}$, and hence is not impacted by the presence of some arms with low reward.\n\nTo estimate $M$, we initialize our estimate as $\\hat{M}=1$, and perform sequential hopping for $2K$ rounds to make sure that every two different players only collide one time. Since players can be orthogonalized thanks to their external rank, a player increments $\\hat{M}$ only if he gets all zero rewards during a specific period (this suggests that some collisions have occurred). As a by-product, we also output the internal rank $j \\in \\{1,...,M\\}$, not to be confused with the external rank $s \\in \\{1,...,K\\}$. Each player is assigned a different internal rank, and it will serve for the last step of our algorithm, in order to assign roles when exploring the various arms.\n\n\\begin{lemma}\\label{lem:virtualnumberplayers}\n Consider $\\delta > 0$. Consider the subroutine \\alg{VirtualNumberPlayers} with input parameters $K$, a good arm $\\tilde{k}$ and $\\tau = {1 \\over \\mu_{\\tilde{k}}} \\ln {1 \\over \\delta}$. Then the subroutine exits after $K^2 \\tau$ time slots causing at most $C_4 K^2 M \\ln {1 \\over \\delta}$ regret, \n and with probability greater than $1 - C_5 \\delta$, it both outputs a correct estimate of the number of players $\\hat{M} = M$, and it assigns a distinct internal rank $j \\in \\{1,...,M\\}$ to all of the $M$ players. Both $C_4$ and $C_5$ are universal constants. \n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{\\alg{VirtualNumberPlayers} (for player $m=1,...,M$)}\n\\label{alg:NumberPlayers}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $\\tilde{k}$ a good arm, $s$: external rank of a player, $\\tau$: sampling times\n\\ENSURE $\\hat{M}$: estimated number of players, $j$: internal rank of the player\n\\STATE $\\hat{M} \\gets 1$, $\\ell \\gets s$, $j \\gets 1$ {\\it \\# initialization}\n\\FOR{$n=1,\\dots,2K$}\n \\ifthen{$n > 2s$}{$\\ell \\gets \\mod{(\\ell+1,K)}$ {\\it \\# sequential hopping}}\n \\STATE $R \\gets 0$ {\\it \\# sum of rewards from the good arm}\n \\STATE {\\it \\# sample from the good arm}\n \\FOR{$k = 1,\\dots,K$}\n \\IF{$\\ell \\ne k$}\n \\for{$t=1,\\dots,\\tau$}{Select an arbitrary arm in $\\{1,...,K\\} \\setminus \\{\\tilde{k}\\}$}\n \\ELSE\n \\STATE {\\it \\# sample from virtual arm $\\tau$ times}\n \\for{$t=1,\\dots,\\tau$}{Select arm $\\tilde{k}$, observe reward $r$, $R \\mathrel{+}= r$} \\STATE {\\it \\# if no non-zero reward was obtained increase the estimated number of players}\n \\IF{$R = 0$}\n \\STATE $\\hat{M} \\mathrel{+}= 1$, \n \\ifthen{$n \\le 2s$}{$j \\mathrel{+}= 1$}\n \\ENDIF\n \\ENDIF\n \\ENDFOR\n\\ENDFOR\n\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Step 4: Distributed Exploration and Finding the Best Arms}\n\nThe last step is the subroutine \\alg{DistributedExploration}, which takes as input $K$ the number of arms, $j$ the internal rank of the player, $\\tau$ the sampling times, $\\hat{M}$ the estimated number of players, and outputs one arm amongst the $M$ best arms, which is assigned to the player. \n\n\\alg{DistributedExploration} is similar to the exploration strategy used in \\cite{shi_ec_sic} \nwith two key improvements. First, to avoid choosing arms with low expected reward, a player sends data only through arm $\\tilde{k}$ rather than through the $j$-th arm as done in \\cite{shi_ec_sic}. Second, we set the quantization error to be dependent on phase p instead of the fixed quantity $\\mu_{M} - \\mu_{M+1}$. By doing so, the number of pulls is reduced, and the subroutine does not require this prior knowledge anymore.\nIts analysis, provided in lemma~\\ref{lem:distributedexploration} follows along similar lines as \\cite{shi_ec_sic}, so we simply highlight the main steps of this procedure. The player with internal rank $j=1$ is called the leader, and other players are called the followers. The procedure operates in phases, and in phase $p$, the active players (the players that have not been assigned an arm yet) sample each active arm (the arms for which players have not figured out whether or not they are amongst the $M$ best arms) $2^p \\lceil \\ln {1 \\over \\delta} \\rceil$ times using sequential hopping (so that no collisions occur). Then all followers send their estimates to the leader, the leader aggregates their estimates, determines which arms can be accepted (the arms for which one is sure that they are amongst the $M$ best arms) and rejected (the arms for which one is sure that they are not amongst the $M$ best arms) and then sends the sets of accepted and rejected arms back to the followers. If an arm is accepted then it is assigned to an active player, and both the arm and the player become inactive, and the player simply selects the assigned arm until the end of the procedure. If an arm is rejected then it becomes inactive. The strategy on how leader and followers act in this information exchange step is given in subroutines \\alg{ComFollow} and \\alg{ComLead} in appendix. In essence, the leader coordinates and takes all of the decisions, while followers simply collect samples, transmit their estimates and receive orders on what to do next. The strategy to send and receive data between leader and followers is given in subroutines \\alg{EncoderSendFloat} and \\alg{DecoderReceiveFloat} in appendix. \n\\begin{lemma}\\label{lem:distributedexploration}\n Consider $\\delta > 0$ and $T \\ge 0$. Consider the subroutine \\alg{DistributedExploration} with input parameters $K$, $\\tilde{k}$ and $\\tau = {1 \\over \\mu_{\\tilde{k}}} \\ln {1 \\over \\delta}$ applied with time horizon $T$ (\\footnote{Applying a procedure with time horizon $T$ simply means that, if it does not terminate before $T$ on its own, then its execution is immediately stopped when time $T$ is reached.}). Then with probability greater that $1 - C_6 \\delta K M (\\ln T)^2 $ the subroutine incurs a regret upper bounded by\n \\begin{align*}\n C_7 & \\sum_{k > M} {\\ln {1 \\over \\delta} \\over \\mu_{(M)} - \\mu_{(k)}} + C_{8} K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln {1 \\over \\delta}\n \\end{align*}\n and assigns one distinct arm to each player from the set of the $M$ best arms, where $C_6,C_7,C_8$ are universal constants.\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{\\alg{DistributedExploration} (for players $m=1,...,M$)}\n\\label{algo}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $j$: internal rank of a player, $\\hat{M}$: number of players, $\\tilde{k}$: a good arm, $\\tau$: sampling times\n\\ENSURE $f$ an arm amongst the $M$ best arms assigned to the player\n\\STATE Initialize $p \\gets 0$; $f \\gets -1$; {\\it \\# initialization}\n\\STATE $R[k], v[k] \\gets 0$ for $k=1,...,K$ {\\it \\# rewards and number of samples for each arm}\n\\STATE {\\it \\# rewards and number of samples for each arm held by each players, only stored by the leader}\n\\IF{$j=1$}\n \\for{$m=1,...,\\hat{M}$ and $k=1,...,K$}{$\\hat\\mu[k,m],N[k,m] \\gets 0$}\n\\ENDIF\n\\STATE $M' \\gets \\hat{M}$, ${\\cal K} \\gets \\{1,\\dots,K\\}$ {\\it \\# number of active players and set of active arms}\n\\WHILE{$f = -1$}\n\\STATE $p \\mathrel{+}= 1$ {\\it \\# start phase $p$}\n\\STATE $k \\gets j$ {\\it \\# first sub-phase explore arms by sequential hopping}\n\\FOR{$t \\gets 1,\\dots, |\\mathcal K| 2^{p} \\left\\lceil \\ln {1 \\over \\delta} \\right\\rceil$}\n\\STATE $k \\gets (k + 1) \\mod |\\mathcal K|$ \n\\STATE Select arm $k$, observe reward $r$, $R[k] \\mathrel{+}= r$, $v[k]\\mathrel{+}= 1$, $E[k] \\gets {R[k] \\over v[k]}$ \n\\ENDFOR\n\\STATE $Q \\gets \\lceil {p \\over 2} + 3 \\rceil$ {\\it \\# second sub-phase: share estimates between players}\n\\IF{$j = 1$}\n\\STATE $(f,{\\cal K},M',\\hat{\\mu},N) \\gets $\\alg{ComLeader}$(\\hat{\\mu},N,{\\cal K},M',Q,\\tau,$ $ \\tilde{k},p,\\delta)$ {\\# player is a leader}\n\\ELSE \n\\STATE $(f,\\mathcal {\\cal K},M') \\gets $\\alg{ComFollow}$(E,j,\\mathcal {\\cal K},M',Q,\\tau,\\tilde{k})$ {\\# player is a follower}\n\\ENDIF\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Putting it all Together}\n\nThe complete proposed algorithm is presented in algorithm~\\ref{alg:proposed}, and combines the four steps above. It is noted that our proposed algorithm is in fact a procedure to identify the set of best arms. To turn it into an algorithm for minimizing regret over a time horizon of $T$, one can simply run the proposed algorithm with confidence parameter $\\delta = {1 \\over T}$, and then let players select the arm that has been assigned to them by the algorithm until the time horizon runs out. The regret upper bound of Theorem~\\ref{th:main} is our main result, and is simply proven by combining the four previous lemmas. As promised, unlike state-of-the-art algorithms, our algorithm does not need any prior information such as the number of arms $K$ or the reward of the worst arm $\\mu_{(K)}$, and its regret does not depend on $\\mu_{(K)}$. The regret bound is better than that of SIC-MMAB2 and EC-SIC, in the sense that the term proportional to $1 \/ \\mu_{(K)}$ (which can be arbitrarily large), has been eliminated. In other words, performance is not limited by the worst arm anymore. This causes a dramatic performance gain which is seen in numerical experiments shown below.\n\n\\begin{theorem}\\label{th:main}\n Consider $T \\ge 0$, $\\mu_{M}-\\mu_{M+1}>0$. First apply the proposed algorithm with input parameters $K$ and $\\delta = {1 \\over T (\\ln T)}$ with time horizon $T$, let $\\bar{k}$ denote its output, and select arm $\\bar{k}$ for the remaining time steps. Then the expected regret of this procedure is \n \\begin{align*}\n R(T) &\\le C_{9} \\sum_{k > M} {\\ln T \\over \\mu_{(M)} - \\mu_{(k)}} + C_{10} K^2 M \\ln T + C_{11} K M^2 \\ln \\left({1 \\over \\mu_{(M)} - \\mu_{(M+1)}}\\right)^2 \\ln T\n \\end{align*}\n with $C_{9},C_{10},C_{11}$ three universal constants.\n\\end{theorem}\n\n\n\\begin{algorithm}\n\\caption{Proposed algorithm (for player $m=1,...,M$)}\n\\label{alg:proposed}\n\\begin{algorithmic}\n\\REQUIRE $K$: number of arms, $\\delta$: confidence level \n\\ENSURE $\\bar{k}$ an arm amongst the $M$ best arms assigned to the player\n\\STATE $(\\tilde{k}, \\tilde{\\mu}) \\gets $\\alg{FindGoodArm}$(K,\\delta)$ {\\it \\# find a good arm and a lower bound on its reward}\n\\STATE $s \\gets $\\alg{VirtualMusicalChairs}$(K, \\tilde{k}, K \\ln(\\frac{1}{\\delta})\/\\tilde{\\mu})$ {\\it \\# assign external rank to each player}\n\\STATE $(\\hat{M},j) \\gets $\\alg{VirtualNumberPlayers}$(K, \\tilde{k}, s, \\ln(\\frac{1}{\\delta}) \/ \\tilde{\\mu})$ {\\it \\# estimate the number of players and assign internal rank}\n\\STATE $\\bar{k} \\gets $\\alg{DistributedExploration}$(K,j,\\hat{M},\\tilde{k}, \\ln(\\frac{1}{\\delta}) \/ \\tilde{\\mu}))$ {\\it \\# find one arms out of the $M$ best arms}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Numerical Experiments}\n\n\nWe now compare the empirical performance of the proposed algorithm against the state-of-the-art algorithms SIC-MMAB2 and EC-SIC. For simplicity we assume that rewards decrease from best to worst in a linear fashion $\\mu_{(k)} = \\mu_{(1)} + {k - 1 \\over K-1} (\\mu_{(K)} - \\mu_{(1)})$. The regret of algorithms is averaged over $20$ (or more) independent runs, and $95\\%$ confidence intervals are presented. \n\n{\\bf Influence of the number of players} In our first set of experiments, we consider $\\mu_{(1)} = 1$ and $\\mu_{(K)} = 0.01$, $M = \\lfloor K\/2 \\rfloor$. We plot the expected regret of the various algorithms for $K = 5,10,20$ in figures \\ref{figureK5M2},\\ref{figureK10M5} and \\ref{figureK20M10} respectively. Overall, the proposed algorithm clearly outperforms EC-SIC and SIC-MMAB2, sometimes by several orders of magnitude, and the difference seems more and more severe when $K$ increases.\n\n{\\bf Influence of the gap} Figure \\ref{figureK5M2} and \\ref{figureK10M5} consider $\\mu_{M}-\\mu_{M+1} = 0.2$ and $\\mu_{M}-\\mu_{M+1} = 0.1$. We see that sometimes the proposed algorithm performs a bit worse than other algorithms for very small time horizons, however for larger time horizons it does outperform SIC-MMAB2 and EC-SIC which seem not to converge quickly enough.\n\n\\begin{figure}[H]\n \\centering\n \\begin{minipage}[t]{0.3\\linewidth}\n \\includegraphics[width=1\\linewidth]{K5M2.pdf}\n \\caption{K=5,M=2,T=$10^5$}\n \\label{figureK5M2}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K10M5.pdf}\n \\caption{K=10,M=5,T=$10^6$}\n \\label{figureK10M5}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K20M10.pdf}\n \\caption{K=20,M=10,T=$10^7$}\n \\label{figureK20M10}\n \\end{minipage}\n\\end{figure}\n\n{\\bf Impact of the number of players} In figures \\ref{figureK10M2} and \\ref{figureK10M8} we compare between $M=2$ players and $M=8$ players. For $M=2$, both the proposed algorithm and EC-SIC do converge. However, when $M$ increases, the collision probability becomes larger, causing EC-SIC to spend significant time performing musical chairs to estimate $M$. \n\\begin{figure}[H]\n \\centering\n \\begin{minipage}[t]{0.3\\linewidth}\n \\includegraphics[width=1\\linewidth]{K10M2.pdf}\n \\caption{K=10,M=2,T=$10^6$}\n \\label{figureK10M2}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K10M8.pdf}\n \\caption{K=10,M=8,T=$10^6$}\n \\label{figureK10M8}\n \\end{minipage}\n \\hspace{0.03\\linewidth}\n \\begin{minipage}[t]{0.3\\linewidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{K5M2vsmu.pdf}\n \\caption{K=5,M=2,T=$10^5$ with different $\\mu_{(K)}$}\n \\label{figureK5M2mu}\n \\end{minipage}\n\\end{figure}\n\n\n{\\bf Impact of the worst arm} We present the regret for different values of $\\mu_{(K)}$ in figure \\ref{figureK5M2mu}. When $\\mu_{(K)}$ decreases, we can see that the performance of SIC-MMAB2 and EC-SIC is greatly affected, and for $\\mu_{(K)} = 0.001$ they do not even start the exploration phase. On the other hand, no matter how small $\\mu_{(K)}$ is, the cumulative regret of the proposed algorithm stays similar, which confirms our theoretical predictions.\n\n\\section{Conclusion}\n\nIn this work we have proposed a new algorithm for multi-player multi-armed bandits without collision sensing information. Through both analysis and numerical experiments, we have proven that it performs significantly better than the state-of-the-art algorithms, while at the same time requiring no input parameter other than the number of arms. The cornerstone of our approach is a novel, non trivial distributed procedure to enable players to discover a good arm without any prior information other than the number of arms. \nWe believe that this result is a significant contribution to the problem as it constitutes a step towards being able to solve multi-player multi-armed bandits in a practical setting (for instance cognitive radios) where no prior information is known about the expected rewards of the arms.\n\n{\\bf Acknowledgement:} The work of Wei Huang was supported by Beyond 5G, a project of the French Government's recovery plan ``France Relance''.\n\n\\newpage \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Preliminaries}\nIn \\cite{bra}, Branciari discussed the existence and uniqueness of fixed points for mappings from a complete metric space $(X,d)$ into itself satisfying a general contractive condition of integral type. The result therein is a generalization of the Banach contraction principle in metric spaces. In fact, Branciari considered mappings $T:(X,d)\\to(X,d)$ satisfying\n$$\\myint{d(Tx,Ty)}\\leq\\alpha\\myint{d(x,y)}\\qquad(x,y\\in X),$$\nwhere $\\alpha\\in(0,1)$ and $\\varphi:[0,+\\infty)\\to[0,+\\infty)$ is a Lebesgue-integrable function on $[0,+\\infty)$ whose Lebesgue-integral is finite on each compact subset of $[0,+\\infty)$, and satisfies $\\myint{\\varepsilon}>0$ for all $\\varepsilon>0$. Recently, an integral version of \\'Ciri\\'c's contraction was given in \\cite{sam}.\\par\nIn 2008, Jachymski \\cite{jac} generalized the Banach contraction principle in metric spaces endowed with a graph. This idea was followed by the authors (see \\cite{knpan,knfpt}) in uniform spaces. In \\cite{aam1}, the concept of an $E$-distance was introduced in uniform spaces as a generalization of a metric and a $w$-distance and then many different nonlinear contractions were generalized from metric to uniform spaces (see, e.g., \\cite{agar,knir,ola}).\\par\nThe aim of this paper is to study the existence and uniqueness of a fixed point for integral type contractions in uniform spaces endowed with both a graph and an $E$-distance. Our results generalize Theorem 2.1 in \\cite{bra} as well as Corollary 3.1 in \\cite{jac} by replacing metric spaces with uniform spaces endowed with a graph and by considering a weaker contractive condition. We also prove an integral version of \\cite[Theorems 3.2 and 3.3]{jac}.\\par\nWe begin with notions in uniform spaces that are needed in this paper. For more detailed discussion, the reader is referred to, e.g., \\cite{wil}.\\par\nBy a uniform space $(X,\\mathscr U)$, shortly denoted here by $X$, it is meant a nonempty set $X$ together with a uniformity $\\mathscr U$. For instance, if $d$ is a metric on a nonempty set $X$, then it induces a uniformity, called the uniformity induced by the metric $d$, in which the members of $\\mathscr U$ are all the supersets of the sets\n$$\\big\\{(x,y)\\in X\\times X:d(x,y)<\\varepsilon\\big\\},$$\nwhere $\\varepsilon>0$.\\par\nIt is well-known that a uniformity $\\mathscr U$ on a nonempty set $X$ is separating if the intersection of all members of $\\mathscr U$ is equal to the diagonal of the Cartesian product $X\\times X$, that is, the set $\\{(x,x):x\\in X\\}$ which is often denoted by $\\Delta(X)$. If $\\mathscr U$ is a separating uniformity on a nonempty set $X$, then the uniform space $X$ is said to be separated.\\par\nWe next recall the definition of an $E$-distance on a uniform space $X$ as well as the notions of convergence, Cauchyness and completeness with $E$-distances.\n\n\\begin{defn}[\\cite{aam1}] Let $X$ be a uniform space. A function $p:X\\times X\\rightarrow[0,+\\infty)$ is called an $E$-distance on $X$ if\n\\begin{enumerate}[label={\\roman*)}]\n\\item for each member $V$ of $\\mathscr U$, there exists a $\\delta>0$ such that $p(z,x)\\leq\\delta$ and $p(z,y)\\leq\\delta$ imply $(x,y)\\in V$ for all $x,y,z\\in X$;\n\\item the triangular inequality holds for $p$, that is,\n$$p(x,y)\\leq p(x,z)+p(z,y)\\qquad(x,y,z\\in X).$$\n\\end{enumerate}\n\\end{defn}\n\nLet $p$ be an $E$-distance on a uniform space $X$. A sequence $\\{x_n\\}$ in $X$ is said to be $p$-convergent to a point $x\\in X$, denoted by $x_n\\stackrel{p}\\longrightarrow x$, if it satisfies the usual metric condition, that is, $p(x_n,x)\\rightarrow0$ as $n\\rightarrow\\infty$, and similarly, $p$-Cauchy if it satisfies $p(x_m,x_n)\\rightarrow0$ as $m,n\\rightarrow\\infty$. The uniform space $X$ is called $p$-complete if every $p$-Cauchy sequence in $X$ is $p$-convergent to some point of $X$.\\par\nIn the next lemma, an important property of $E$-distances in separated uniform spaces is formulated.\n\n\\begin{lem}[\\cite{aam1}]\\label{1}\nLet $p$ be an $E$-distance on a separated uniform space $X$ and $\\{x_n\\}$ and $\\{y_n\\}$ be two arbitrary sequences in $X$. If $x_n\\stackrel{p}\\longrightarrow x$ and $x_n\\stackrel{p}\\longrightarrow y$, then $x=y$. In particular, if $x,y\\in X$ and $p(z,x)=p(z,y)=0$ for some $z\\in X$, then $x=y$.\n\\end{lem}\n\nFinally, we recall some concepts about graphs. For more details on graph theory, see, e.g., \\cite{bon}.\\par\nLet $X$ be a uniform space and consider a directed graph $G$ without any parallel edges such that the set $V(G)$ of its vertices is $X$, that is, $V(G)=X$ and the set $E(G)$ of its edges contains all loops, that is, $E(G)\\supseteq\\Delta(X)$. So the graph $G$ can be simply denoted by $G=(V(G),E(G))$. By $\\widetilde G$, it is meant the undirected graph obtained from $G$ by ignoring the direction of the edges of $G$, that is,\n$$V(\\widetilde G)=X\\quad\\text{and}\\quad E(\\widetilde G)=\\big\\{(x,y)\\in X\\times X:\\text{either}\\ (x,y)\\ \\text{or}\\ (y,x)\\ \\text{belongs to}\\ E(G)\\big\\}.$$\\par\nA subgraph $H$ of $G$ is itself a directed graph such that $V(H)$ and $E(H)$ are contained in $V(G)$ and $E(G)$, respectively, and $(x,y)\\in E(H)$ implies $x,y\\in V(H)$ for all $x,y\\in X$.\\par\nWe need also a few notions about connectivity of graphs. Suppose that $x$ and $y$ are two vertices in $V(G)$. A finite sequence $(x_i)_{i=0}^N$ consisting of $N+1$ vertices of $G$ is a path in $G$ from $x$ to $y$ if $x_0=x$, $x_N=y$ and $(x_{i-1},x_i)\\in E(G)$ for $i=1,\\ldots,N$. The graph $G$ is weakly connected if there exists a path in $\\widetilde G$ between each two vertices of $\\widetilde G$.\n\n\n\\section{Main Results}\nIn this section, we consider the Euclidean metric on $[0,+\\infty)$ and denote by $\\lambda$ the Lebesgue measure on the Borel $\\sigma$-algebra of $[0,+\\infty)$. For a Borel set $E=[a,b]$, we will use the notation $\\int_a^b\\varphi(t){\\rm d}t$ to show the Lebesgue integral of a function $\\varphi$ on $E$. We employ a class $\\Phi$ consisting of all functions $\\varphi:[0,+\\infty)\\rightarrow[0,+\\infty)$ satisfying the following properties:\n\\begin{enumerate}[label={$(\\Phi\\arabic*)$}]\n\\item $\\varphi$ is Lebesgue-integrable on $[0,+\\infty)$;\n\\item The value of the Lebesgue integral $\\myint{\\varepsilon}$ is positive and finite for all $\\varepsilon>0.$\n\\end{enumerate}\\par\nThe next lemma embodies some important properties of functions of the class $\\Phi$ which we need in the sequel.\n\n\\begin{lem}\\label{phi}\nLet $\\varphi:[0,+\\infty)\\to[0,+\\infty)$ be a function in the class $\\Phi$ and $\\{a_n\\}$ be a sequence of nonnegative real numbers. Then the following statements hold:\n\\begin{enumerate}[label={\\rm\\arabic*.}]\n\\item If $\\myint{a_n}\\rightarrow0$ as $n\\rightarrow\\infty$, then $a_n\\rightarrow0$ as $n\\rightarrow\\infty$.\n\\item If $\\{a_n\\}$ is monotone and converges to some $a\\geq0$, then $\\myint{a_n}\\rightarrow\\myint a$ as $n\\rightarrow\\infty$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n1. Let $\\myint{a_n}\\rightarrow0$ and suppose first on the contrary that $\\limsup_{n\\rightarrow\\infty}a_n=\\infty$. Then $\\{a_n\\}$ contains a subsequence $\\{a_{n_k}\\}$ which diverges to $\\infty$. By passing to a subsequence if necessary, one may assume without loss of generality that $\\{a_{n_k}\\}$ is a nondecreasing subsequence of $\\{a_n\\}$. Because the sequence $\\{\\myint{a_{n_k}}\\}$ of nonnegative numbers increases to zero, so $a_{n_k}=0$ for all $k\\geq1$. This is a contradiction and therefore the sequence $\\{a_n\\}$ is bounded.\\par\nNext, if $\\limsup_{n\\rightarrow\\infty}a_n=\\varepsilon>0$, then there exists a strictly increasing sequence $\\{n_k\\}$ of positive integers such that $a_{n_k}\\rightarrow\\varepsilon$. Pick an integer $k_0>0$ so that the strict inequality $a_{n_k}>\\frac\\varepsilon2$ holds for all $k\\geq k_0$. Therefore,\n$$0<\\myint{\\frac\\varepsilon2}\\leq\\myint{a_{n_k}}\\rightarrow0,$$\nwhich is again a contradiction. So $\\limsup_{n\\rightarrow\\infty}a_n=0$, and consequently,\n$$0\\leq\\liminf_{n\\rightarrow\\infty}a_n\\leq\\limsup_{n\\rightarrow\\infty}a_n=0,$$\nthat is, $a_n\\rightarrow0$.\\\\\n2. Let $\\{a_n\\}$ be nondecreasing and put $E_n=[0,a_n]$ for all $n\\geq1$. Then each $E_n$ is a Borel subset of $[0,+\\infty)$ and we have $E_1\\subseteq E_2\\subseteq\\cdots$ and $\\bigcup_{n=1}^\\infty E_n=[0,a]$. Because the function $E\\stackrel{\\mu}\\longmapsto\\int_E\\varphi{\\rm d}\\lambda$ is a Borel measure on $[0,+\\infty)$, using the continuity of $\\mu$ from below we get\n$$\\myint a=\\mu\\Big(\\bigcup_{n=1}^\\infty E_n\\Big)=\\lim_{n\\rightarrow\\infty}\\mu(E_n)=\\lim_{n\\rightarrow\\infty}\\myint{a_n}.$$\nA similar argument is true if $\\{a_n\\}$ is nonincreasing since each $E_n$ defined above is of finite $\\mu$-measure by $(\\Phi2)$.\n\\end{proof}\n\nLet $T$ be a mapping from a uniform space $X$ endowed with a graph $G$ into itself. We denote as usual the set of all fixed points for $T$ by $\\fix(T)$, and by $C_T$, we mean the set of all $x\\in X$ such that $(T^nx,T^mx)$ is an edge of $\\widetilde G$ for all $m,n\\geq0$. Clearly, $\\fix(T)\\subseteq C_T$.\n\n\\begin{defn}\nLet $p$ be an $E$-distance on a uniform space $X$ endowed with a graph $G$. We say that a mapping $T:X\\to X$ is an integral type $p$-$G$-contraction if\n\\begin{enumerate}[label={IC\\arabic*)}]\n\\item $T$ preserves the egdes of $G$, that is, $(x,y)\\in E(G)$ implies $(Tx,Ty)\\in E(G)$ for all $x,y\\in X$;\n\\item there exists a $\\varphi\\in\\Phi$ and a constant $\\alpha\\in(0,1)$ such that the contractive condition\n$$\\myint{p(Tx,Ty)}\\leq\\alpha\\myint{p(x,y)}$$\nholds for all $x,y\\in X$ with $(x,y)\\in E(G)$.\n\\end{enumerate}\n\\end{defn}\n\nNow, we give some examples of integral type $p$-$G$-contractions.\n\n\\begin{exm}\nLet $p$ be an $E$-distance on a uniform space $X$ endowed with a graph $G$ and $x_0$ be a point in $X$ such that $p(x_0,x_0)=0$. Since $E(G)$ contains the loop $(x_0,x_0)$, it follows that the constant mapping $T=x_0$ preserves the edges of $G$, and since $p(x_0,x_0)=0$, (IC2) holds trivially for any arbitrary $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$. Therefore, $T$ is an integral type $p$-$G$-contraction. In particular, each constant mapping on $X$ is an integral type $p$-$G$-contraction if and only if $p(x,x)=0$ for all $x\\in X$.\n\\end{exm}\n\n\\begin{exm}\nLet $(X,d)$ be a metric space and $T:X\\to X$ a mapping satisfying\n$$\\myint{d(Tx,Ty)}\\leq\\alpha\\myint{d(x,y)}\\qquad(x,y\\in X),$$\nwhere $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$. If we consider $X$ as a uniform space with the uniformity induced by the metric $d$, then $T$ is an integral type $d$-$G_0$-contraction, where $G_0$ is the complete graph with the vertices set $X$, that is, $V(G_0)=X$ and $E(G_0)=X\\times X$. The existence and uniqueness of fixed point for these kind of contractions were considered by Branciari in \\cite{bra}.\n\\end{exm}\n\n\\begin{exm}\nLet $\\preceq$ and $p$ be a partial order and an $E$-distance on a uniform space $X$, respectively, and consider the poset graphs $G_1$ and $G_2$ by\n$$V(G_1)=X\\quad\\text{and}\\quad E(G_1)=\\big\\{(x,y)\\in X\\times X:x\\preceq y\\big\\},$$\nand\n$$V(G_2)=X\\quad\\text{and}\\quad E(G_2)=\\big\\{(x,y)\\in X\\times X:x\\preceq y\\vee y\\preceq x\\big\\}.$$\nThen integral type $p$-$G_1$-contractions are precisely the ordered integral type $p$-contractions, that is, nondecreasing mappings $T:X\\to X$ which satisfy (IC2) for all $x,y\\in X$ with $x\\preceq y$ and for some $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$. And integral type $p$-$G_2$-contractions are those mappings $T:X\\to X$ which are order preserving and satisfy (IC2) for all comparable $x,y\\in X$ and for some $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$.\n\\end{exm}\n\n\\begin{rem}\nLet $T$ be a mapping from an arbitrary uniform space $X$ into itself. If $X$ is endowed with the complete graph $G_0$, then the set $C_T$ coincides with $X$.\\par\nIf $\\preceq$ is a partial order on $X$ and $X$ is endowed with either $G_1$ or $G_2$, then a point $x\\in X$ belongs to $C_T$ if and only if $T^nx$ is comparable to $T^mx$ for all $m,n\\geq0$. In particular, if $T$ is monotone, then each $x\\in X$ satisfying $x\\preceq Tx$ or $Tx\\preceq x$ belongs to $C_T$.\n\\end{rem}\n\n\\begin{exm}\nLet $p$ be any arbitrary $E$-distance on a uniform space $X$ endowed with a graph $G$ and define a function $\\varphi:[0,+\\infty)\\to[0,+\\infty)$ by the rule $\\varphi(t)=t^\\beta$ for all $t\\geq0$, where $\\beta\\geq0$ is constant. It is clear that $\\varphi$ is Lebesgue-integrable on $[0,+\\infty)$ and $\\myint{\\varepsilon}=\\frac{\\varepsilon^{1+\\beta}}{1+\\beta}$ which is positive and finite for all $\\varepsilon>0$, that is, $\\varphi\\in\\Phi$. Now, let a mapping $T:X\\to X$ satisfy $p(Tx,Ty)\\leq\\alpha p(x,y)$ for all $x,y\\in X$ with $(x,y)\\in E(G)$, where $\\alpha\\in(0,1)$. Then $T$ satisfies (IC2) for the function $\\varphi$ defined as above and the number $\\alpha^{1+\\beta}\\in(0,1)$. In fact, if $x,y\\in X$ and $(x,y)\\in E(G)$, then\n$$\\myint{p(Tx,Ty)}=\\frac{p(Tx,Ty)^{1+\\beta}}{1+\\beta}\\leq\\alpha^{1+\\beta}\\cdot\\frac{p(x,y)^{1+\\beta}}{1+\\beta}=\\alpha^{1+\\beta}\\myint{p(x,y)}.$$\nTherefore, our contraction generalizes Banach's contraction with $E$-distances in uncountably many ways. In particular,\nif $T$ is a Banach $G$-$p$-contraction (i.e., the Banach contraction in uniform spaces endowed with an $E$-distance and a graph), then $T$ is an integral type $p$-$G$-contraction for uncountably many functions $\\varphi\\in\\Phi$.\n\\end{exm}\n\nTo prove the existence of a fixed point for an integral type $p$-$\\widetilde G$-contraction, we need the following two lemmas:\n\n\\begin{lem}\\label{step1}\nLet $p$ be an $E$-distance on a uniform space $X$ endowed with a graph $G$ and $T:X\\to X$ be an integral type $p$-$G$-contraction. Then $p(T^nx,T^ny)\\rightarrow0$ as $n\\rightarrow\\infty$, for all $x,y\\in X$ with $(x,y)\\in E(G)$.\n\\end{lem}\n\n\\begin{proof}\nLet $x,y\\in X$ be such that $(x,y)\\in E(G)$. According to Lemma \\ref{phi}, it suffices to show that $\\myint{p(T^nx,T^ny)}\\rightarrow0$, where $\\varphi\\in\\Phi$ is as in (IC2). To this end, note that because $T$ preserves the edges of $G$, we have $(T^nx,T^ny)\\in E(G)$ for all $n\\geq0$, and so by (IC2), we find\n$$\\myint{p(T^nx,T^ny)}\\leq\\alpha\\myint{p(T^{n-1}x,T^{n-1}y)}\\leq\\cdots\\leq\\alpha^n\\myint{p(x,y)}\\qquad(n\\geq1),$$\nwhere $\\alpha\\in(0,1)$ is as in (IC2). Since, by $(\\Phi2)$, $\\myint{p(x,y)}$ is finite (even possibly zero), it follows immediately that $\\myint{p(T^nx,T^ny)}\\rightarrow0$.\n\\end{proof}\n\n\\begin{lem}\\label{step2}\nLet $p$ be an $E$-distance on a uniform space $X$ endowed with a graph $G$ and $T:X\\to X$ be an integral type $p$-$\\widetilde G$-contraction. Then the sequence $\\{T^nx\\}$ is $p$-Cauchy for all $x\\in C_T$.\n\\end{lem}\n\n\\begin{proof}\nSuppose on the contrary that $\\{T^nx\\}$ is not $p$-Cauchy for some $x\\in C_T$. Then there exist an $\\varepsilon>0$ and positive integers $m_k$ and $n_k$ such that\n$$m_k>n_k\\geq k\\quad\\text{and}\\quad p(T^{m_k}x,T^{n_k}x)\\geq\\varepsilon\\qquad k=1,2,\\ldots\\,.$$\nIf the integer $n_k$ is kept fixed for sufficiently large indices $k$ (say, $k\\geq k_0$), then using Lemma \\ref{step1}, one may assume without loss of generality that $m_k>n_k$ is the smallest integer with $p(T^{m_k}x,T^{n_k}x)\\geq\\varepsilon$, that is,\n$$p(T^{m_k-1}x,T^{n_k}x)<\\varepsilon\\qquad(k\\geq k_0).$$\nHence we have\n\\begin{eqnarray*}\n\\varepsilon&\\leq&p(T^{m_k}x,T^{n_k}x)\\cr\n&\\leq&p(T^{m_k}x,T^{m_k-1}x)+p(T^{m_k-1}x,T^{n_k}x)\\cr\n&<&p(T^{m_k}x,T^{m_k-1}x)+\\varepsilon\n\\end{eqnarray*}\nfor each $k\\geq k_0$. Since $x\\in C_T$, it follows that $(Tx,x)\\in E(\\widetilde G)$ and by Lemma \\ref{step1}, we have $p(T^{m_k}x,T^{m_k-1}x)\\rightarrow0$. Thus, letting $k\\rightarrow\\infty$ yields $p(T^{m_k}x,T^{n_k}x)\\rightarrow\\varepsilon$. On the other hand, we have\n$$p(T^{m_k+1}x,T^{n_k+1}x)\\leq p(T^{m_k+1}x,T^{m_k}x)+p(T^{m_k}x,T^{n_k}x)+p(T^{n_k}x,T^{n_k+1}x)$$\nfor all $k\\geq1$. Letting $k\\rightarrow\\infty$, since $(Tx,x),(x,Tx)\\in E(\\widetilde G)$, it follows by Lemma \\ref{step1} that\n$$\\limsup_{k\\rightarrow\\infty}p(T^{m_k+1}x,T^{n_k+1}x)\\leq\\varepsilon.$$\nMoreover, the inequality\n$$p(T^{m_k+1}x,T^{n_k+1}x)\\geq p(T^{m_k}x,T^{n_k}x)-p(T^{m_k}x,T^{m_k+1}x)-p(T^{n_k+1}x,T^{n_k}x)$$\nholds for all $k\\geq1$. Thus, similarly we have\n$$\\liminf_{k\\rightarrow\\infty}p(T^{m_k+1}x,T^{n_k+1}x)\\geq\\varepsilon.$$\nHence, $p(T^{m_k+1}x,T^{n_k+1}x)\\rightarrow\\varepsilon$. By passing to two subsequences with the same choice function if necessary, one may assume without loss of generality that both $\\{p(T^{m_k}x,T^{n_k}x)\\}$ and $\\{p(T^{m_k+1}x,T^{n_k+1}x)\\}$ are monotone. Therefore, using Lemma \\ref{phi} twice, we have\n$$\\myint{\\varepsilon}=\\lim_{k\\rightarrow\\infty}\\myint{p(T^{m_k+1}x,T^{n_k+1}x)} \\leq\\alpha\\lim_{k\\rightarrow\\infty}\\myint{p(T^{m_k}x,T^{n_k}x)}=\\alpha\\myint{\\varepsilon},$$\nwhere $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$ are as in (IC2). Therefore, $\\myint{\\varepsilon}=0$, which is a contradiction. Consequently, the sequence $\\{T^nx\\}$ is $p$-Cauchy for all $x\\in C_T$.\n\\end{proof}\n\n\\begin{defn}\nLet $p$ be an $E$-distance on a uniform space $X$ endowed with a graph $G$ and $T$ be a mapping from $X$ into itself. We say that\n\\begin{enumerate}[label={\\roman*)}]\n\\item $T$ is orbitally $p$-$G$-continuous on $X$ if for all $x,y\\in X$ and all sequences $\\{a_n\\}$ of positive integers with $(T^{a_n}x,T^{a_{n+1}}x)\\in E(G)$ for $n=1,2,\\ldots$, $T^{a_n}x\\stackrel{p}\\longrightarrow y$ as $n\\rightarrow\\infty$, implies $T(T^{a_n}x)\\stackrel{p}\\longrightarrow Ty$ as $n\\rightarrow\\infty$.\n\\item $T$ is a $p$-Picard operator if $T$ has a unique fixed point $u\\in X$ and $T^nx\\stackrel{p}\\longrightarrow u$ for all $x\\in X$.\n\\item $T$ is a weakly $p$-Picard operator if $\\{T^nx\\}$ is $p$-convergent to a fixed point of $T$ for all $x\\in X$.\n\\end{enumerate}\n\\end{defn}\n\nIt is clear that each $p$-Picard operator is weakly $p$-Picard. Moreover, a weakly $p$-Picard operator is $p$-Picard if and only if its fixed point is unique.\n\n\\begin{exm}\nLet $X$ be any arbitrary uniform space with more than one point equipped with an $E$-distance $p$. Choose a nonempty proper subset $A$ of $X$ and pick $a$ and $b$ from $A$ and $A^c$, respectively. Then the mapping $T:X\\to X$ defined by $Tx=a$ if $x\\in A$, and $Tx=b$ if $x\\notin A$ is a weakly $p$-Picard operator which fails to be $p$-Picard. In fact, we have $\\fix(T)=\\{a,b\\}$. Therefore, a weakly $p$-Picard operator is not necessarily $p$-Picard.\n\\end{exm}\n\nNow, we are ready to prove our main theorems. The first result guarantees the existence of a fixed point when an integral type $p$-$\\widetilde G$-contraction is orbitally $p$-$\\widetilde G$-continuous on $X$ or the triple $(X,p,G)$ has a certain property.\n\n\\begin{thm}\\label{contfix}\nLet $p$ be an $E$-distance on a separated uniform space $X$ endowed with a graph $G$ such that $X$ is $p$-complete, and $T:X\\to X$ be an integral type $p$-$\\widetilde G$-contraction. Then $T\\mid_{C_T}$ is a weakly $p$-Picard operator if one of the following statements holds:\n\\begin{enumerate}[label={\\rm\\roman*)}]\n\\item $T$ is orbitally $p$-$\\widetilde G$-continuous on $X$;\n\\item The triple $(X,p,G)$ satisfies the following property:\n\\begin{itemize}[label={$(\\ast)$}]\n\\item If a sequence $\\{x_n\\}$ in $X$ is $p$-convergent to an $x\\in X$ and satisfies $(x_n,x_{n+1})\\in E(\\widetilde G)$ for all $n\\geq1$, then there exists a subsequence $\\{x_{n_k}\\}$ of $\\{x_n\\}$ such that $(x_{n_k},x)\\in E(\\widetilde G)$ for all $k\\geq1$.\n\\end{itemize}\n\\end{enumerate}\nIn particular, having been held {\\rm(i)} or {\\rm(ii)}, $\\fix(T)\\ne\\emptyset$ if and only if $C_T\\ne\\emptyset$.\n\\end{thm}\n\n\\begin{proof}\nIf $C_T=\\emptyset$, then there is nothing to prove. Otherwise, note first that since $T$ preserves the edges of $\\widetilde G$, it follows that $C_T$ is $T$-invariant, that is, $T$ maps $C_T$ into itself. Now, let $x\\in C_T$ be given. Then $(T^nx,T^{n+1}x)\\in E(\\widetilde G)$ for all $n\\geq0$. Moreover, by Lemma \\ref{step2}, the sequence $\\{T^nx\\}$ is $p$-Cauchy in $X$, and because $X$ is $p$-complete, there exists a $u\\in X$ (depends on $x$) such that $T^nx\\stackrel{p}\\longrightarrow u$.\\par\nTo prove the existence of a fixed point for $T$, suppose first that $T$ is orbitally $p$-$\\widetilde G$-continuous. Then $T^{n+1}x\\stackrel{p}\\longrightarrow Tu$ and because $X$ is separated, Lemma \\ref{1} ensures that $Tu=u$, that is, $u$ is a fixed point for $T$.\\par\nOn the other hand, if Property $(\\ast)$ holds, then $\\{T^nx\\}$ contains a subsequence $\\{T^{n_k}x\\}$ such that $(T^{n_k}x,u)\\in E(\\widetilde G)$ for all $k\\geq1$. Since $p(T^{n_k}x,u)\\rightarrow0$, by passing to a subsequence if necessary, one may assume without loss of generality that $\\{p(T^{n_k}x,u)\\}$ is monotone. Hence by Lemma \\ref{phi}, we have\n$$\\myint{p(T^{{n_k}+1}x,Tu)}\\leq\\alpha\\myint{p(T^{n_k}x,u)}\\rightarrow0\\quad\\text{as}\\quad k\\rightarrow\\infty,$$\nwhere $\\alpha\\in(0,1)$ is as in (IC2). Using Lemma \\ref{phi} once more, one obtains $p(T^{{n_k}+1}x,Tu)\\rightarrow0$ and since $X$ is separated, Lemma \\ref{1} guarantees that $Tu=u$, that is, $u$ is a fixed point for $T$.\\par\nFinally, $u\\in\\fix(T)\\subseteq C_T$, and so $T\\mid_{C_T}$ is a weakly $p$-Picard operator.\n\\end{proof}\n\nSetting $G=G_0$ in Theorem \\ref{contfix}, we have the following result, which is a generalization of \\cite[Theorem 2.1]{bra} to uniform spaces equipped with an $E$-distance.\n\n\\begin{cor}\nLet $p$ be an $E$-distance on a separated uniform space $X$ such that $X$ is $p$-complete. Let $T:X\\to X$ satisfy\n$$\\myint{p(Tx,Ty)}\\leq\\alpha\\myint{p(x,y)}\\qquad(x,y\\in X),$$\nwhere $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$. Then $T$ is a $p$-Picard operator.\n\\end{cor}\n\n\\begin{proof}\nBy Theorem \\ref{contfix}, the mapping $T$ is a weakly $p$-Picard operator. To complete the proof, it suffices to show that $T$ has a unique fixed point. To this end, let $x$ and $y$ be two fixed points for $T$. Then\n$$\\myint{p(x,y)}=\\myint{p(Tx,Ty)}\\leq\\alpha\\myint{p(x,y)},$$\nwhich is impossible unless $p(x,y)=0$. Similarly, one can show that $p(x,x)=0$ and since $X$ is separated, it follows by Lemma \\ref{1} that $x=y$.\n\\end{proof}\n\nBecause $\\widetilde{G_1}=\\widetilde{G_2}=G_2$, setting $G=G_1$ or $G=G_2$ in Theorem \\ref{contfix}, we obtain the ordered version of Branciari's result as follows:\n\n\\begin{cor}\nLet $p$ be an $E$-distance on a partially ordered separated uniform space $X$ such that $X$ is $p$-complete and a mapping $T:X\\to X$ satisfy\n$$\\myint{p(Tx,Ty)}\\leq\\alpha\\myint{p(x,y)}$$\nfor all comparable elements $x$ and $y$ of $X$, where $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$. Assume that there exists an $x\\in X$ such that $T^mx$ and $T^nx$ are comparable for all $m,n\\geq0$. Then $T$ is a weakly $p$-Picard operator if one of the following statements holds:\n\\begin{itemize}[label={$-$}]\n\\item $T$ is orbitally $p$-$G_2$-continuous on $X$;\n\\item $X$ satisfies the following property:\n\\begin{itemize}\n\\item[] If a sequence $\\{x_n\\}$ in $X$ with successive comparable terms is $p$-convergent to an $x\\in X$, then $x$ is comparable to $x_n$ for all $n\\geq1$.\n\\end{itemize}\n\\end{itemize}\n\\end{cor}\n\nNext, we are going to prove two theorems on uniqueness of the fixed points for integral type $p$-$\\widetilde G$-contractions.\n\n\\begin{thm}\nLet $p$ be an $E$-distance on a separated uniform space $X$ endowed with a graph $G$ such that $X$ is $p$-complete, and let $T:X\\to X$ be an integral type $p$-$\\widetilde G$-contraction such that the function $\\varphi$ in (IC2) satisfies\n\\begin{equation}\\label{subadd}\n\\myint{a+b}\\leq\\myint{a}+\\myint{b}\n\\end{equation}\nfor all $a,b\\geq0$. If $G$ is weakly connected and $C_T$ is nonempty, then there exists a unique $u\\in X$ such that $T^nx\\stackrel{p}\\longrightarrow u$ for all $x\\in X$. In particular, $T$ is a $p$-Picard operator if and only if $\\fix(T)$ is nonempty.\n\\end{thm}\n\n\\begin{proof}\nLet $x$ and $y$ be two arbitrary elements of $X$. Since $G$ is weakly connected, there exists a path $(x_i)_{i=0}^N$ in $\\widetilde G$ from $x$ to $y$. Since $T$ preserves the edges of $\\widetilde G$, it follows that $(T^nx_{i-1},T^nx_i)\\in E(\\widetilde G)$ for all $n\\geq0$ and $i=1,\\ldots,N$. Therefore, by \\eqref{subadd} and (IC2) we have\n\\begin{eqnarray*}\n\\myint{p(T^nx,T^ny)}&\\leq&\\myint{\\sum_{i=1}^Np(T^nx_{i-1},T^nx_i)}\\cr\\\\\n&\\leq&\\sum_{i=1}^N\\myint{p(T^nx_{i-1},T^nx_i)}\\cr\\\\\n&\\leq&\\alpha\\sum_{i=1}^N\\myint{p(T^{n-1}x_{i-1},T^{n-1}x_i)}\\cr\\\\\n&\\vdots&\\cr\\\\[-5mm]\n&\\leq&\\alpha^n\\sum_{i=1}^N\\myint{p(x_{i-1},x_i)}\n\\end{eqnarray*}\nfor all $n\\geq0$, where $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$ are as in (IC2). Since, by $(\\Phi2)$, $\\sum_{i=1}^N\\myint{p(x_{i-1},x_i)}$ is finite (possibly zero), it follows immediately that $\\myint{p(T^nx,T^ny)}\\rightarrow0$. Hence by Lemma \\ref{phi}, $p(T^nx,T^ny)\\rightarrow0$.\\par\nNow, pick a point $x\\in C_T$. By Lemma \\ref{step2}, the sequence $\\{T^nx\\}$ is $p$-Cauchy in $X$ and since $X$ is $p$-complete, there exists a $u\\in X$ such that $T^nx\\stackrel{p}\\longrightarrow u$. If $y$ is an arbitrary point in $X$, then\n$$0\\leq p(T^ny,u)\\leq p(T^ny,T^nx)+p(T^nx,u)\\rightarrow0\\quad\\text{as}\\quad n\\rightarrow\\infty.$$\nSo $T^ny\\stackrel{p}\\longrightarrow u$. The uniqueness of $u$ follows immediately from Lemma \\ref{1}.\n\\end{proof}\n\n\\begin{thm}\\label{thm4}\nLet $p$ be an $E$-distance on a separated uniform space $X$ endowed with a graph $G$ and $T:X\\to X$ be an integral type $p$-$\\widetilde G$-contraction. If the subgraph of $G$ with the vertices $\\fix(T)$ is weakly connected, then $T$ has at most one fixed point in $X$.\n\\end{thm}\n\n\\begin{proof}\nLet $x$ and $y$ be two fixed points for $T$. Then there exists a path $(x_i)_{i=0}^N$ in $\\widetilde G$ from $x$ to $y$ such that $x_1,\\ldots,x_{N-1}\\in\\fix(T)$. Since $E(\\widetilde G)$ contains all loops, we can assume without loss of generality that the length of this path, that is, the integer $N$ is even. Now, by (IC2) we have\n$$\\myint{p(x_{i-1},x_i)}=\\myint{p(Tx_{i-1},Tx_i)}\\leq\\alpha\\myint{p(x_{i-1},x_i)}\\qquad i=1,\\ldots,N,$$\nwhere $\\varphi\\in\\Phi$ and $\\alpha\\in(0,1)$, which is impossible unless $\\myint{p(x_{i-1},x_i)}=0$ or equivalently, $p(x_{i-1},x_i)=0$ for $i=1,\\ldots,N$. Because $E(\\widetilde G)$ is symmetric, a similar argument yields $p(x_i,x_{i-1})=0$ for $i=1,\\ldots,N$. Since $N$ is even, using Lemma \\ref{1} finitely many times, we get $x=x_0=x_2=\\cdots=x_N=y$. Consequently, $T$ has at most one fixed point in $X$.\n\\end{proof}\n\n\\begin{rem}\nTheorem \\ref{thm4} guarantees that in a separated uniform space $X$ endowed with a graph $G$ and an $E$-distance $p$, if $(x,y)\\in E(G)$, then both $x$ and $y$ cannot be a fixed point for any integral type $p$-$\\widetilde G$-contraction $T$. In other words, each weakly connected component of $G$ intersects $\\fix(T)$ in at most one point. So in partially ordered separated uniform spaces equipped with an $E$-distance $p$, no ordered integral type $p$-contraction has two comparable fixed points.\n\\end{rem}\n\n\\begin{rem}\nSince the Riemann integral (proper and improper) is subsumed in the Lebesgue integral, it follows that one may replace Lebesgue-integrability with Riemann-integrability of $\\varphi$ on $[0,+\\infty)$ in $(\\Phi1)$, where the value of the integral on $[0,+\\infty)$ is allowed to be $\\infty$. Facing with Riemann integrals, we should assume that the function $\\varphi$ is bounded. Therefore, all of the results of this paper can be restated and reproved for Riemann integrals instead of Lebesgue integrals. A similar remark holds for Riemann-Stieltjes integrable functions with respect to any fixed nondecreasing function on $[0,+\\infty)$.\n\\end{rem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nProbing the response of a nucleus to the scattering of a particle or photon is a powerful tool to study the underlying nuclear structure. In the excitation energy range from 10 MeV to 30 MeV the nuclear systems show prominent and broad resonances, that are called giant resonances. Giant resonances have been experimentally studied for a long time \\cite{Harakeh2001}, and yet the techniques that are developed \nare still improving towards unprecedented and advanced levels \\cite{Gales2018,Bracco2019}. These studies provide extremely rich information\non the nuclear phenomenology. To name a few highlights, we mention the study of compression modes such as the Isoscalar Giant Monopole and Dipole Resonances that are undertaken in order to understand the incompressibility of uniform nuclear matter \\cite{Garg2018}; the Isovector Giant Dipole Resonance and the associated dipole polarizability that is studied due to its implication for the symmetry energy \\cite{Trippa2008,Roca-Maza2013,Roca-Maza2015,Bracco2019}; the low-lying dipole strength in the Isovector Dipole channel for its possible relation to the neutron skin thickness \\cite{Wieland2011,Roca-Maza2012a,Savran2013,Bracco2015,Burrello2018}; the Isoscalar Giant Quadrupole Resonance which is tightly connected to the nucleon effective mass close to the Fermi surface \\cite{Roca-Maza2013a}; the Gamow-Teller resonance \\cite{Fujita2011} for its key role in astrophysically relevant weak-interaction processes \\cite{Langanke2003} (cf. also the general discussion about giant resonances and the parameters of the nuclear equation of state in Ref.~\\cite{Roca-Maza2018a}, and the report on giant resonances in nuclei far from stability \nin Ref.~\\cite{Paar2007}).\n\nOn the other hand, the rich information on giant resonances also sets a challenge for theoretical descriptions. In the random phase approximation (RPA), the giant resonances are described as coherent superpositions of one particle one hole (1p-1h) excitations. The centroid of giant resonances and the energy weighted sum rule (EWSR) can be well described. However, the experimental resonance width $\\Gamma$, which directly relates with the \nlifetime $\\tau \\equiv \\frac{\\hbar}{\\Gamma}$, cannot be described by RPA due to the missing of the coupling with more complicated correlations.\n\nTwo main effects were identified to contribute to the width, the escape of a nucleon from the nucleus (escape width) and the spreading of the excitation energy into more complicated configurations (spreading width) \\cite{Bertsch1983}.\nDifferent efforts have been made to take into account these effects, for example, in second RPA (SRPA) the coupling of 1p-1h excitations with two particle two hole (2p-2h) excitations is taken into account \\cite{Papakonstantinou2009,Papakonstantinou2010,Gambacurta2010,Gambacurta2018,Vasseur2018}.\nIn quasiparticle-phonon model, the excited states are composed of two-phonon excitations \\cite{Soloviev1992,VanGiai1998,Iudice2012,Severyukhin2018}.\nThe quasiparticle-phonon model based on time-blocking approximation (TBA) \\cite{Tselyaev2007,Lyutorovich2018a,Tselyaev2018} and its relativistic extension (RTBA) \\cite{Litvinova2007,Litvinova2018,Robin2018} are developed within the many-body Green's function formalism, in which the 1p-1h (or two quasiparticles) $\\otimes$ phonon configurations are included and more complicated intermediate states are blocked.\nIn the equation-of-motion phonon method, a set of equations for multiphonon states are derived, within the Tamm-Dancoff approximation (TDA).\n\nRecently, a systematic approach to the response functions with nonperturbative treatment of higher configurations is formulated in the equation-of-motion framework, with a truncation at the level of two-fermion correlation functions.\nThe PVC-TBA method is then compared here with the formulas of the equation-of-motion approach and this provides a guidance to develop a systematic treatment of the response functions \\cite{Litvinova2018a}.\n\nIn this work we use the particle-vibration coupling (PVC) model, which takes into account the coupling between a nucleon and the low-lying nuclear collective excitations (phonons) \\cite{Bohr1998b}. In early applications, phenomenological inputs were used for the PVC vertex and parameters were adjusted to reproduce the data, making it difficult to have a universal description \\cite{Bortignon1977,Bertsch1983,Mahaux1985}. A self-consistent treatment for the interaction in the PVC vertex, on top of a mean field associated with Skyrme functionals was worked out in Ref.~\\cite{Bernard1980}, although only the velocity-independent central term was included in the vertex. The approach was further developed in Refs.~\\cite{Colo1992,Colo1994,Colo2010,CaoLG2014}, and now the full Skyrme interaction is used for both the PVC vertex and the mean field. The same consistency has been achieved also in the relativistic PVC \\cite{Litvinova2006}.\nThe PVC has been extended to describe open-shell nuclei within the Hartree-Fock (HF) plus BCS framework \\cite{Colo2001}, and later on in the Hartree-Fock-Bogoliubov framework \\cite{NiuYF2016}. By including both collective and noncollective excitations, the so-called hybrid configuration mixing (HCM) model was developed to study the low-lying spectroscopy of odd nuclei, and shell-model-type states like 2p-1h can be well taken into account \\cite{Colo2017}. To better understand the renormalization of the effective interaction, the subtraction method developed in Ref.~\\cite{Tselyaev2007} has been studied in the PVC \\cite{Roca-Maza2017}. The PVC has been used to investigate, for example, the $\\beta$-decay \\cite{NiuYF2015,NiuYF2018} and good descriptions were achieved.\n\nHowever, in the above PVC studies the so-called diagonal approximation has been used, that is, the 1p-1h state coupled with a phonon, which is also called a doorway state, has no interaction with other doorway states. This is similar to the diagonal approximation in the SRPA where there is no interaction among the 2p-2h states.\nIn the context of SRPA, this approximation has been tested against the fully self-consistent framework, and it has been shown to affect significantly the strength distributions \\cite{Gambacurta2010}.\nSuch correlations have also been considered in the TBA \\cite{Tselyaev2007} and studied by RTBA \\cite{Litvinova2010,Litvinova2013}, based on the equation-of-motion method \\cite{Schuck1976}.\nIt was found that the two-phonon correlations push large part of the pygmy strength above the neutron threshold, in better agreement with available data for the tin isotopes and $^{68}$Ni \\cite{Litvinova2010,Litvinova2013}.\nFurther progresses on higher-order correlations beyond the 2p-2h level of configuration can be found in Refs~\\cite{Tselyaev2018,Litvinova2015}.\n\nTherefore, it is of importance to have a closer view into the diagonal approximation in the PVC.\nHere we adopt a different approach from the works by Refs. \\cite{Litvinova2010,Litvinova2013}.\nWe will use the equation-of-motion method similar to the one used in the SRPA described in Ref.~\\cite{Yannouleas1987}, and the two particle-holes inside the doorway states will interact through the particle-hole interaction.\nWe will compare the effect of removing diagonal approximation with the ones in SRPA \\cite{Gambacurta2010} and RTBA \\cite{Litvinova2010,Litvinova2013}.\nAn analytical comparison to the formalisms of RTBA will also be given in the Appendix.\nThe sum rules in current PVC framework will also be discussed both analytically and numerically.\n\nIn Sec. \\ref{sec:theory}, we give a brief summary of the formalisms of the HF, RPA, and PVC. The numerical details for the calculations are discussed in Sec. \\ref{sec:nd}. Results for the isoscalar giant monopole, dipole, and quadrupole resonances of $^{16}$O by PVC without diagonal approximation are presented in Sec. \\ref{sec:res}. Finally, the summary and perspectives for future investigations will be given in Sec. \\ref{sec:sum}.\n\n\n\\section{Theoretical Framework}\\label{sec:theory}\n\n\\subsection{From Hartree-Fock to Random Phase Approximation}\n\nOur starting point is the Skyrme functional which is constructed from the Skyrme effective interaction solved within the Hartree-Fock (HF) approximation. The detail of the Skyrme interaction and the corresponding formulas of the Skyrme Hartree-Fock theory in spherical nuclei have been given in detail \\cite{Vautherin1972} and will not be repeated here. In this work we take the doubly magic nucleus $^{16}$O as an example, so that effects of pairing and deformation \\cite{Vautherin1973} can be ignored. The Hartree-Fock ground state $|\\Phi_0^{\\rm HF}\\rangle$ is a single Slater determinant.\nIn the second quantized form it can be written as:\n\\begin{equation}\\label{eq:}\n |\\Phi_0^{\\rm HF}\\rangle = \\prod_i^A a_i^\\dagger |\\rangle,\n\\end{equation}\nwhere $A$ is the number of nucleons of a given nucleus, $a_i^\\dagger$ is the creation operator of HF single-particle state $|i\\rangle$, and $|\\rangle$ is the bare vacuum. The HF equation is solved with a box boundary condition and a set of discrete occupied and unoccupied states $|i\\rangle$ are obtained. The Hamiltonian of the system can be expressed as\n\\begin{equation}\\label{eq:}\n H = H_0 + V_{\\rm res},\n\\end{equation}\nwhere $H_0$ is the HF Hamiltonian and $V_{\\rm res}$ the residual interaction:\n\\begin{align}\n H_0 &= \\sum_i^A e_i a_i^\\dagger a_i^{} - \\frac{1}{2} \\sum_{ij}^A \\bar{V}_{ijij}, \\\\\n V_{\\rm res} &= \\frac{1}{4} \\sum_{k'l'kl} \\bar{V}_{k'l'kl} :a_{k'}^\\dagger a_{l'}^\\dagger a_l^{} a_k^{}:.\n\\label{eq:}\n\\end{align}\nIn the above equations, $e_i$ is the single-particle energy of state $|i\\rangle$, and $\\bar{V}_{ijij} = V_{ijij} - V_{ijji}$ is the antisymmetrized two-body matrix element. The normal ordered product of operators $a_{k'}^\\dagger a_{l'}^\\dagger a_l^{} a_k^{}$ is labelled as $:a_{k'}^\\dagger a_{l'}^\\dagger a_l^{} a_k^{}:$ with respect to the HF particle-hole vacuum $|\\Phi_0^{\\rm HF}\\rangle$.\n\nTo study the excited state properties, one can use the RPA, in which all the possible 1p-1h excitations are considered. If we define the HF ground state $|\\Phi_0^{\\rm HF}\\rangle$ and all the 1p-1h excitations $|ph\\rangle$ built upon as the subspace $Q_1$, the RPA solution can be obtained by diagonalizing the Hamiltonian in this subspace $Q_1HQ_1$. For the derivation of the RPA equations and their solution we refer the reader to Ref. \\cite{Ring1980}.\nThe RPA equation reads\n\\begin{equation}\\label{eq:rpa}\n \\sum_{ph} \\left(\\begin{array}{cc}\n A & B \\\\ -B^* & -A^*\n \\end{array}\\right)_{p'h',ph}\n \\left(\\begin{array}{c}\n X_{ph}^{(n)} \\\\ Y_{ph}^{(n)}\n \\end{array}\\right)\n = \\omega_n\n \\left(\\begin{array}{c}\n X_{p'h'}^{(n)} \\\\ Y_{p'h'}^{(n)}\n \\end{array}\\right)\n\\end{equation}\nwith $\\omega_n$ the excitation energy of RPA state $|\\Phi_n^{\\rm RPA}\\rangle$ (that can be simply labeled as $|n\\rangle$ when there is no ambiguity), $X_{ph}^{(n)}$ and $Y_{ph}^{(n)}$ the corresponding RPA wave function coefficients. The matrix elements $A$ and $B$ are\n\\begin{subequations}\\label{eq:ab}\\begin{align}\n A_{p'h',ph} &= \\langle 0|[a_{h'}^\\dagger a_{p'}^{}, [H, a_{p}^\\dagger a_{h}^{} ]] |0 \\rangle, \\notag \\\\\n &= \\delta_{p'h',ph} (e_p - e_h) + \\bar{V}_{p'hh'p}, \\\\\n B_{p'h',ph} &= -\\langle 0|[a_{h'}^\\dagger a_{p'}^{}, [H, a_{h}^\\dagger a_{p}^{} ]] |0 \\rangle = \\bar{V}_{p'ph'h},\n\\end{align}\\end{subequations}\nwhere $|0\\rangle$ is the RPA ground state $|\\Phi_0^{\\rm RPA}\\rangle$, and within the quasiboson approximation it is replaced by the HF ground state $|\\Phi_0^{\\rm HF}\\rangle$ \\cite{Ring1980}. Without causing confusion, the simple form $|0\\rangle$ of the ground state will be used later on also in the framework of PVC. The RPA excited states, or the phonons, can be expressed as\n\\begin{equation}\\label{eq:}\n |n\\rangle = Q_n^\\dagger |0\\rangle,\n\\end{equation}\nwith\n\\begin{equation}\\label{eq:qn}\n Q_n^\\dagger = \\sum_{ph} \\left[ X_{ph}^{(n)} a_p^\\dagger a_h^{} - Y_{ph}^{(n)} a_h^\\dagger a_p^{} \\right],\n\\end{equation}\nand the RPA ground state satisfies\n\\begin{equation}\\label{eq:}\n Q_n |0\\rangle \\equiv 0.\n\\end{equation}\n\n\\subsection{Particle-vibration coupling}\n\nAs we briefly mentioned in the Introduction,\nRPA can give a good description of the centroid energy of giant resonances as well as of the EWSR exhausted by each mode.\nHowever, properties such as the width of the resonances cannot be well described. Part of the width comes from the so called Landau damping effect and part of it is due to correlations beyond 1p-1h \\cite{Bertsch1983}. The Landau damping effect \nproduces a fragmentation of the strength, in contrast with the ideal situation in which there is a single collective peak. Such an effect depends on the intensity of the residual interaction that 1p-1h configurations feel, as well as the density of the unperturbed 1p-1h states around the \nresonance energy.\nCoupling with more complicated states than 1p-1h produce the resonance spreading width.\nOur formalism can also account for the other mechanism giving rise to the resonance width, since the escape of a nucleon can be also described. \n\nTo take into account these effects, two subspaces $P$ and $Q_2$ are built. Similar to $Q_1$, subspace $P$ is made up with 1p-1h configurations but now the particle is a continuum state and orthogonal to all the states in $|i\\rangle$. For subspace $Q_2$, one can chose the 2p-2h configurations and the resulting framework would be the second RPA \\cite{Yannouleas1983}. In the particle-vibration coupling model, the $Q_2$ space is composed of the so-called doorway states $|N\\rangle$ with 1p-1h excitation coupled to a RPA phonon,\n\\begin{equation}\\label{eq:doorway}\n |N\\rangle = |ph\\rangle \\otimes |n\\rangle.\n\\end{equation}\nThe corresponding excitation operator reads\n\\begin{equation}\\label{eq:QN}\n \\tilde{Q}_N^\\dagger = \\sum_{ph,n} \\left[ \\tilde{X}_{ph,n}^{N} a_p^\\dagger a_h^{} Q_n^\\dagger - \\tilde{Y}_{ph,n}^{N} Q_n^{} a_h^\\dagger a_p^{} \\right].\n\\end{equation}\n\nNow, the PVC equation is an eigenequation in the $P+Q_1+Q_2$ space,\n\\begin{equation}\\label{eq:}\n H (P+Q_1+Q_2)\\Psi = \\omega (P+Q_1+Q_2)\\Psi,\n\\end{equation}\n$\\Psi$ being the full-space wave function to be projected out.\nAfter truncating higher orders, this equation can be mapped into $Q_1$ with an energy dependent Hamiltonian as \\cite{Colo1994} (see Appendix \\ref{app:hq1})\n\\begin{equation}\\label{eq:pvceq}\n \\mathcal{H}(\\omega) Q_1\\Psi = \\left( \\Omega_\\nu - i\\frac{\\Gamma_\\nu}{2} \\right) Q_1\\Psi.\n\\end{equation}\nBoth the effective Hamiltonian $\\mathcal{H}$ and the eigensolutions are complex.\nThe effective Hamiltonian is composed of three terms,\n\\begin{widetext}\n\\begin{equation}\\label{eq:homega}\n \\mathcal{H}(\\omega) \\equiv Q_1HQ_1 + W^\\uparrow(\\omega) + W^\\downarrow(\\omega)\n = Q_1HQ_1 + Q_1HP\\frac{1}{\\omega-PHP+i\\epsilon} PHQ_1 + Q_1HQ_2\\frac{1}{\\omega-Q_2HQ_2+i\\epsilon} Q_2HQ_1,\n\\end{equation}\n\\end{widetext}\ni.e., the RPA term, escape term ($W^\\uparrow$), and spreading term ($W^\\downarrow$).\nFor the calculation of the escape term, one is referred to Ref. \\cite{Colo1994}.\nFor more detail of the spreading term and the diagonal approximation of it, see Section \\ref{sec:spread}.\n\nAs one is now working in the $Q_1$ subspace, the RPA solutions can be used as a basis to expand the PVC state as\n\\begin{equation}\\label{eq:wf}\n |\\nu\\rangle = \\sum_n F_n^{(\\nu)} |n\\rangle.\n\\end{equation}\nThen the PVC equation (\\ref{eq:pvceq}) takes the matrix form\n\\begin{equation}\\label{eq:pvceq2}\n \\sum_n \\mathcal{H}_{n'n}(\\omega) F_{n}^{(\\nu)} = \\left( \\Omega_\\nu - i\\frac{\\Gamma_\\nu}{2} \\right) F_{n'}^{(\\nu)},\n\\end{equation}\nwith\n\\begin{equation}\\label{eq:hnn}\n \\mathcal{H}_{n'n}(\\omega) = \\omega_n + W_{n'n}^\\uparrow(\\omega) + W_{n'n}^\\downarrow(\\omega).\n\\end{equation}\nThe matrix of the wave function coefficients is complex orthogonal,\n\\begin{equation}\\label{eq:}\n F^TF = FF^T = 1.\n\\end{equation}\n\nThe polarizability associated with the operator $O$ is defined as\n\\begin{equation}\\label{eq:polar}\n \\Pi(\\omega) = \\langle 0| O^\\dagger \\frac{1}{\\omega-\\mathcal{H}(\\omega)+i\\epsilon}O|0 \\rangle.\n\\end{equation}\nThe corresponding strength function is\n\\begin{align}\n S(\\omega) &= -\\frac{1}{\\pi} {\\rm Im} \\Pi(\\omega) \\notag \\\\\n &= -\\frac{1}{\\pi} {\\rm Im}\\sum_\\nu \\langle 0|O|\\nu \\rangle^2 \\frac{1}{\\omega-\\Omega_\\nu+i\\frac{\\Gamma_\\nu}{2}}.\n\\label{eq:str}\n\\end{align}\nThe sum rules, or the $k$th moments $m_k$ of the strength function, are defined as\n\\begin{equation}\\label{eq:mk}\n m_k = \\int_0^\\infty S(\\omega) \\omega^k d\\omega.\n\\end{equation}\nAmong them, the energy-weighted sum rule $m_1$ is of particular interest as it can be expressed in a simple form via a double commutator evaluated in the ground state, namely\n\\begin{equation}\\label{eq:dbc}\n m_1 = \\frac{1}{2} \\langle 0|[O^\\dagger,[H,O]]|0 \\rangle.\n\\end{equation}\n\n\\subsection{Spreading term in PVC}\\label{sec:spread}\n\nThe spreading term is the last term in Eq.~(\\ref{eq:homega}),\n\\begin{equation}\\label{eq:wd}\n W^\\downarrow(\\omega) = Q_1HQ_2\\frac{1}{\\omega-Q_2HQ_2+i\\epsilon} Q_2HQ_1.\n\\end{equation}\nIt describes the process in which 1p-1h configurations of the $Q_1$ subspace are coupled to the more complicated doorway states of the $Q_2$ subspace.\nThese terms can be derived with the equation-of-motion method \\cite{Rowe1968} as in the SRPA~\\cite{Yannouleas1987}.\nSimilar to the RPA matrix $Q_1HQ_1$ in Eq.~(\\ref{eq:rpa}), one has the matrix $Q_1HQ_2$ and $Q_2HQ_2$ in the particle-hole and phonon representation:\n\\begin{align}\n Q_1HQ_2 &=\n \\left(\\begin{array}{cc}\n A_{ph,p_1h_1n} & B_{ph,p_1h_1n} \\\\\n -B_{ph,p_1h_1n}^* & -A_{ph,p_1h_1n}^* \\\\\n \\end{array}\\right) \\label{eq:q1hq2} \\\\\n Q_2HQ_2 &=\n \\left(\\begin{array}{cc}\n A_{p_1h_1n_1,p_2h_2n_2} & B_{p_1h_1n_1,p_2h_2n_2} \\\\\n -B_{p_1h_1n_1,p_2h_2n_2}^* & -A_{p_1h_1n_1,p_2h_2n_2}^* \\\\\n \\end{array}\\right) \n\\label{eq:q2hq2}\n\\end{align}\nwith the matrix elements defined similarly to Eq.~(\\ref{eq:ab}),\n\\begin{align}\n A_{ph,p_1h_1n} &= \\langle 0|[a_h^\\dagger a_p^{}, [H, a_{p_1}^\\dagger a_{h_1}^{} Q_n^\\dagger ]] |0 \\rangle, \\\\\n B_{ph,p_1h_1n} &= -\\langle 0|[a_h^\\dagger a_p^{}, [H, Q_n^{} a_{h_1}^\\dagger a_{p_1}^{} ]] |0 \\rangle, \\\\\n A_{p_1h_1n_1,p_2h_2n_2} &= \\langle 0|[Q_{n_1}^{} a_{h_1}^\\dagger a_{p_1}^{}, [H, a_{p_2}^\\dagger a_{h_2}^{} Q_{n_2}^\\dagger ]] |0 \\rangle, \\\\\n B_{p_1h_1n_1,p_2h_2n_2} &= -\\langle 0|[Q_{n_1}^{} a_{h_1}^\\dagger a_{p_1}^{}, [H, Q_{n_2}^{} a_{h_2}^\\dagger a_{p_2}^{} ]] |0 \\rangle.\n\\label{eq:b22}\n\\end{align}\nThey can be evaluated as\n\\begin{align}\n A_{ph,p_1h_1n} &= \\delta_{hh_1} \\langle p|V|p_1,n \\rangle - \\delta_{pp_1} \\langle h_1|V|h,n \\rangle, \\label{eq:a12} \\\\\n A_{p_1h_1n_1,p_2h_2n_2} &= \\delta_{n_1n_2} \\left[ \\delta_{p_1h_1,p_2h_2} \\left( \\omega_{n_1} + e_{p_1h_1} \\right) + \\bar{V}_{p_1h_2h_1p_2} \\right], \\label{eq:a22} \\\\\n B_{ph,p_1h_1n} &= B_{p_1h_1n_1,p_2h_2n_2} = 0, \\label{eq:b12}\n\\end{align}\nwith $\\omega_n$ the energy of the phonon $|n\\rangle$, $e_{ph} = e_p - e_h$, and\n\\begin{equation}\\label{eq:vabn}\n \\langle a|V|b,n \\rangle = \\sum_{ph} \\left[ X_{ph}^{(n)} \\bar{V}_{ahbp} + Y_{ph}^{(n)} \\bar{V}_{apbh} \\right].\n\\end{equation}\n\nThe matrix element $A_{ph,p_1h_1n}$ in Eq.~(\\ref{eq:a12}) represents the interaction between the 1p-1h state $|ph\\rangle$ in the $Q_1$ space and the doorway state $|p_1h_1\\rangle\\otimes|n\\rangle$ in the $Q_2$ space.\nA diagrammatic representation of this interaction is given in the left part of Fig.~\\ref{fig:q1hq2}, where straight lines are denoted for fermions (with up-arrow a particle and down-arrow a hole), red wave lines are for phonons. The solid circle between two particle (or two hole) lines and a phonon is for the phonon vertex $\\langle p|V|p1,n \\rangle$ (or $\\langle h1|V|h,n \\rangle$) in Eq.~(\\ref{eq:a12}).\nThe matrix element $A_{p_1h_1n_1,p_2h_2n_2}$ in Eq.~(\\ref{eq:a22}) represents the interaction among the doorway states, \nand its diagrammatic representation is also provided in the right part of Fig.~\\ref{fig:q1hq2}.\nThe non-interacting part (first one) is denoted as $\\delta_{n_1n_2} \\delta_{p_1h_1,p_2h_2} \\left( \\omega_{n_1} + e_{p_1h_1} \\right)$, and the dashed line in the interacting part (second one) is for the interaction between two particle-holes $\\bar{V}_{p_1h_2h_1p_2}$ in Eq.~(\\ref{eq:a22}).\n\n\\begin{figure}[h]\n \\includegraphics[width=8cm]{vertex}\n \\caption{Schematic picture of the interaction of $Q_1HQ_2$ and $Q_2HQ_2$, corresponding to the matrix elements in Eqs.~(\\ref{eq:a12}-\\ref{eq:a22}). The straight lines are used to represent fermions (with up-arrow for a particle and down-arrow for a hole), while wave lines are for phonon states. The solid circle between two particle (or two hole) lines and a phonon is for the phonon vertex $\\langle p|V|p1,n \\rangle$ (or $\\langle h1|V|h,n \\rangle$) in Eq.~(\\ref{eq:a12}), the dashed line is for the interaction between two particle-holes $\\bar{V}_{p_1h_2h_1p_2}$ in Eq.~(\\ref{eq:a22}).}\n \\label{fig:q1hq2}\n\\end{figure}\n\nThe full spreading term can then be written as\n\\begin{align}\n W_{p'h',ph}^\\downarrow(\\omega) &= \\sum_{p_1'h_1'p_1h_1n} A_{p'h',p_1'h_1'n} \\notag \\\\\n &\\times \\left( \\omega-A_{p_1'h_1'n,p_1h_1n} +i\\epsilon\\right)^{-1}\n A_{p_1h_1n,ph}.\n\\label{eq:wph}\n\\end{align}\nIn the above notation $\\left( \\omega-A_{p_1'h_1'n,p_1h_1n} +i\\epsilon\\right)^{-1}$ is not the inverse of a single matrix element, but the matrix element of the inverted matrix of $\\omega-A_{p_1'h_1'n,p_1h_1n} +i\\epsilon$. \n\nIn previous investigations, the diagonal approximation was used, that is, no interaction among the doorway states was considered \\cite{Colo1994}.\nWithin this approximation, the matrix element $A_{p_1h_1n_1,p_2h_2n_2}$ in Eq.~(\\ref{eq:a22}) becomes\n\\begin{equation}\\label{eq:a22dia}\n A_{p_1h_1n_1,p_2h_2n_2} = \\delta_{n_1n_2} \\delta_{p_1h_1,p_2h_2} \\left( \\omega_{n_1} + e_{p_1h_1} \\right).\n\\end{equation}\nThe matrix $Q_2HQ_2$ then becomes diagonal, and the spreading term can be easily evaluated as\n\\begin{equation}\\label{eq:wph2}\n W_{p'h',ph}^\\downarrow(\\omega) = \\sum_{p_1h_1n} \\frac{A_{ph,p_1h_1n}A_{p_1h_1n,ph}}{\\omega-\\omega_n-e_{p_1h_1}+i\\epsilon}.\n\\end{equation}\n\nWhen the diagonal approximation is not considered, there is an extra step of inverting the matrix $\\omega - Q_2HQ_2 + i\\epsilon$ before evaluating the spreading term.\nSee Appendix \\ref{app:me} for more details.\n\nFinally, interactions that are fitted at the mean-field level and are used within effective \ntheories that go beyond mean field should in principle be refitted against to experimental data in order to avoid double-counting. That is, a renormalization of the model parameters is compulsory. The parameters will change their value since many-body contributions beyond mean-field are now explicitly included. The purpose of the subtraction method \\cite{Tselyaev2007,Roca-Maza2017} is to provide a recipe for the renormalization of the effective interaction within the adopted model scheme that avoids a refitting of the parameters. For that, the spreading term in Eq.~(\\ref{eq:homega}) should be replaced by\n\\begin{equation}\\label{eq:sub}\n W^\\downarrow(\\omega) \\to W^\\downarrow(\\omega) - W^\\downarrow(\\omega = 0).\n\\end{equation}\n\n\n\\subsection{Sum rules}\\label{sec:sr}\n\nIn this subsection we discuss the sum rules.\nFollowing a similar derivation from the response theory in the extended RPA \\cite{Adachi1988}, the sum rules (\\ref{eq:mk}) can be obtained as\n\\begin{equation}\\label{eq:}\n m_k = \\frac{1}{2} O^\\dagger \\mathscr{T} (\\mathscr{M}^{-1} \\mathscr{I})^k \\mathscr{M}^{-1} \\mathscr{T}^\\dagger O,\n\\end{equation}\nwith $O$ the one-body excitation operator same as in Eq.~(\\ref{eq:polar}),\n\\begin{equation}\\label{eq:oij}\n O = \\sum_{ij} O_{ij} a_i^\\dagger a_j.\n\\end{equation}\n$\\mathscr{I}$ is the stability matrix, and $\\mathscr{T}$ and $\\mathscr{M}$ are the metric matrices,\n\\begin{align}\n \\mathscr{T}_{c\\beta} &=\n \\left(\\begin{array}{cc}\n \\tilde{U}_{c\\beta} & \\tilde{V}_{c\\beta}\n \\end{array}\\right), \\quad\n \\mathscr{M}_{\\alpha\\beta} =\n \\left(\\begin{array}{cc}\n {U}_{\\alpha\\beta} & {V}_{\\alpha\\beta} \\\\\n -{V}_{\\alpha\\beta} & -{U}_{\\alpha\\beta}\n \\end{array}\\right).\n\\label{eq:metric}\n\\end{align}\nFor $Q_1$ subspaces only, considering the case of $p'h'$ ($h'p'$ will be similar) one has\n\\begin{align}\n \\tilde{U}_{c\\beta} &: \\tilde{U}_{p'h',ph} = \\langle 0|[a_{h'}^\\dagger a_{p'}, a_{p}^\\dagger a_h]|0 \\rangle = \\delta_{p'h',ph}, \\\\\n \\tilde{V}_{c\\beta} &: \\tilde{V}_{p'h',ph} = \\langle 0|[a_{h'}^\\dagger a_{p'}, a_{h}^\\dagger a_p]|0 \\rangle = 0, \\\\\n {U}_{\\alpha\\beta} &: {U}_{p'h',ph} = \\langle 0|[a_{h'}^\\dagger a_{p'}, a_{p}^\\dagger a_h]|0 \\rangle = \\tilde{U}_{p'h',ph}, \\\\\n {V}_{\\alpha\\beta} &: {V}_{p'h',ph} = \\langle 0|[a_{h'}^\\dagger a_{p'}, a_{h}^\\dagger a_p]|0 \\rangle = \\tilde{V}_{p'h',ph}.\n\\label{eq:}\n\\end{align}\nThe index $c$ is used to denote the pair $a_i^\\dagger a_j$ in Eq. (\\ref{eq:oij}).\nThe indices $\\alpha$ and $\\beta$ for $Q_1$ subspace are for $a_p^\\dagger a_h$ or $a_h^\\dagger a_p$ in Eq.~(\\ref{eq:qn}); for $Q_2$ subspace they are for $a_p^\\dagger a_h Q_n^\\dagger$ and $Q_na_h^\\dagger a_p$ in Eq.~(\\ref{eq:QN}); for $P$ subspace it is similar to $Q_1$ but with the particle in the continuum, which we will label as\n\\begin{equation}\\label{eq:qtilden}\n Q_{\\tilde{n}}^\\dagger = \\sum_{\\tilde{p}h} \\left[ X_{\\tilde{p}h}^{(\\tilde{n})} a_{\\tilde{p}}^\\dagger a_h - Y_{\\tilde{p}h}^{(\\tilde{n})} a_h^\\dagger a_{\\tilde{p}} \\right].\n\\end{equation}\nWhen $P$ and $Q_2$ subspaces are included, the dimension of metric matrices in Eq.~(\\ref{eq:metric}) will be enlarged accordingly.\nIt has been shown for SRPA \\cite{Adachi1988} that even the 2p-2h correlations are considered in the excitation state, only the 1p1h components of $(\\mathscr{M}^{-1} \\mathscr{I})^k$ contribute to the sum rules, because of the absence of ground-state correlations.\nThis can be seen when one tries to evaluate the metric matrix elements with the ground state $|0\\rangle$ chosen as the HF state $|\\Phi_0^{\\rm HF}\\rangle$,\n\\begin{equation}\\label{eq:}\n \\tilde{U}_{ij,p_1p_2h_1h_2} = \\langle 0|[a_{j}^\\dagger a_{i}, a_{p_1}^\\dagger a_{p_2}^\\dagger a_{h_2}a_{h_1}]|0 \\rangle = 0.\n\\end{equation}\nIn the end one can prove that the $m_0$ and $m_1$ are the same for SRPA and RPA \\cite{Adachi1988}.\nFor extended RPA, however, the 2p-2h correlations are also included in the ground state $|0\\rangle$ and in this case $\\tilde{U}_{ij,p_1p_2h_1h_2}$ has non-zero components, pp ($\\tilde{U}_{pp,p_1p_2h_1h_2}$) and hh ($\\tilde{U}_{hh,p_1p_2h_1h_2}$).\nAs a result, $m_0$ and $m_1$ are different from RPA \\cite{Adachi1988}.\n\nIn our PVC framework, the $Q_2$ subspace (\\ref{eq:doorway},\\ref{eq:QN}) is similar to the 2p-2h subspace in SRPA, and the ground state $|0\\rangle$ is also chosen as the HF ground state.\nIt is then not difficult to find a similar conclusion as in SRPA,\n\\begin{equation}\\label{eq:}\n \\tilde{U}_{ij,phn} = \\langle 0|[a_{j}^\\dagger a_{i}, a_{p}^\\dagger a_{h} Q_{n}^\\dagger]|0 \\rangle = 0,\n\\end{equation}\nthat is, the one-body excitation operator $O$ (\\ref{eq:oij}) cannot connect the ground state to the $Q_2$ subspace in our framework.\nTherefore, as a result, the $m_0$ and $m_1$ should be the same as in RPA.\nThis also agrees with the TBA, that when the effective interaction coincides with the one of RPA, the EWSR is the same as RPA \\cite{Tselyaev2007}.\n\nFor the $P$ subspace (\\ref{eq:qtilden}), the following term in the metric matrix is non-zero\n\\begin{equation}\\label{eq:}\n \\tilde{U}_{\\tilde{p}'h',\\tilde{p}h} = \\langle 0|[a_{h'}^\\dagger a_{\\tilde{p}'}, a_{\\tilde{p}}^\\dagger a_{h}]|0 \\rangle = \\delta_{\\tilde{p}'h',\\tilde{p}h}.\n\\end{equation}\nWhile such contribution from the continuum ($P$ subspace) should be small, the approximations done in dealing with the escape term $W^\\uparrow$ in Eq. (\\ref{eq:homega}) (see, e.g., Ref. \\cite{Colo1994}) could make an influence and in the end the sum rules given by PVC with the escape term could be slightly different from those of RPA.\nThe numerical results will be shown in Sec. \\ref{sec:res}.\n\nIn any case, when the diagonal approximation is removed, the sum rules $m_0$ and $m_1$ will not be influenced as this approximation only affect the interaction $Q_2HQ_2$.\nSimilar to SRPA, this part will affect the sum rules from $m_3$, which is \\cite{Adachi1988}\n\\begin{align}\n m_3^{\\rm SRPA} &= \\frac{1}{2} O^\\dagger H_{11}^3\\mathscr{M} O\n + \\frac{1}{2} O^\\dagger \\left( H_{12}H_{21}H_{11} \\right. \\notag \\\\\n &~~~ \\left. + H_{11}H_{12}H_{21} + H_{12}H_{22}H_{21} \\right) \\mathscr{M} O,\n\\label{eq:}\n\\end{align}\nwith expressions $H_{11} = Q_1HQ_1$ and so on.\n\n\n\\section{Numerical details}\\label{sec:nd}\n\nThe nucleus $^{16}$O is studied as an example since it provides a simple case for various theoretical investigations and tests.\nAs it is a doubly magic nucleus, the effects of pairing and deformation can be ignored.\nThe Skyrme functional SAMi \\cite{Roca-Maza2012} will be used in all calculations except in the last section where a systematic study on the dependence on the parameterization of the Skyrme functional is given.\nThree isoscalar (IS) non charge-exchange excitation modes will be examined: the giant monopole resonance (GMR, $J^\\pi = 0^+$), giant dipole resonance (GDR, $J^\\pi = 1^-$), and giant quadrupole resonance (GQR, $J^\\pi = 2^+$).\nThe corresponding adopted excitation operators are \\cite{Colo2013}\n\\begin{subequations}\\label{eq:}\\begin{align}\n O({\\rm ISGMR}) &= \\sum_{i=1}^A r_i^2 Y_{00}, \\\\\n O({\\rm ISGDR}) &= \\sum_{i=1}^A \\left( r_i^3 - \\frac{5\\langle r^2 \\rangle}{3} r_i \\right) Y_{1M}, \\\\\n O({\\rm ISGQR}) &= \\sum_{i=1}^A r_i^2 Y_{2M},\n\\end{align}\\end{subequations}\nwith $r_i$ the radial coordinate of the $i$'th nucleon and $Y_{LM}$ the spherical harmonic function.\nThe special form of the ISGDR is aimed at removing the contribution from the spurious state \\cite{Colo2013}.\nThe spurious state in the RPA solution has also been excluded in the selection of doorway states $|N\\rangle$ in Eq.~(\\ref{eq:doorway}).\nThe corresponding EWSR is evaluated by the double commutator (DC) in Eq.~(\\ref{eq:dbc}) with HF ground state $|0\\rangle = |\\Phi_0^{\\rm HF}\\rangle$ \\cite{Colo2013}:\n\\begin{subequations}\\label{eq:m1dc}\\begin{align}\n m_1^{\\rm (DC)}({\\rm ISGMR}) &= \\frac{\\hbar^2}{2m} \\frac{A}{\\pi} \\langle r^2 \\rangle, \\\\\n m_1^{\\rm (DC)}({\\rm ISGDR}) &= \\frac{\\hbar^2}{2m} \\frac{A}{4\\pi} \\left( 33\\langle r^4 \\rangle - 25\\langle r^2 \\rangle^2 \\right), \\\\\n m_1^{\\rm (DC)}({\\rm ISGQR}) &= \\frac{\\hbar^2}{2m} \\frac{25A}{2\\pi} \\langle r^2 \\rangle,\n\\end{align}\\end{subequations}\nwith $m$ the nucleon mass.\nTo take into account the 1-body center-of-mass correction, in the end the DC sum rules are to be multiplied by a factor of $(A-1)\/A$.\n\n\\begin{figure*}[htbp!]\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16mon.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16isdip.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16qua.eps}\n \\hspace{-0.2cm}\n \\caption{(Color online) Strength function of ISGMR (a), ISGDR (b), and ISGQR (c) in $^{16}$O calculated by RPA with full interaction, PVC with diagonal approximation and central interaction (PVC-dia, $V_c$), with diagonal approximation and full interaction (PVC-dia, $V_{\\rm full}$), without diagonal approximation and with full interaction (PVC, $V_{\\rm full}$).\n In all cases the SAMi functional is used.\n See text for the detail of the experimental data \\cite{Harakeh1981,Lui2001}.}\n \\label{fig:str}\n\\end{figure*}\n\nThe HF equation is solved in a spherical box with size $R = 20$ fm and a radial step $dr = 0.1$ fm.\nIn the RPA calculation, the single-particle energy cut-off is $e_{\\rm cut} = 80$ MeV, so that it ensures the convergence of our results as it can be seen from the EWSR for ISGMR, ISGDR, and ISGQR in $^{16}$O that are all $100\\%$ fulfilled, see the column ``RPA'' in Table~\\ref{tab:sr}.\nFor the PVC calculation, the phonons selected in the doorway states, i.e., the summation index $n$ in Eq.~(\\ref{eq:wph}), include multipolarity $J^\\pi = 0^+, 1^-, 2^+, 3^-, 4^+, 5^-$.\nContributions from unnatural parity states such as $0^-$ should be negligible and therefore are not included.\nConvergence of the results by considering natural parity phonons up to $5^-$ is well achieved.\nThe phonon energy cut-off is $\\omega_{n,{\\rm cut}} = 30$ MeV.\nA further criteria for the selection of phonons is its strength, only those phonons with $B(EJ)\/m_0 \\geq F_{\\rm cut}$ will be selected in the doorway states and the fraction cut-off is $F_{\\rm cut} = 2\\%$.\nThese cut-offs have been checked in previous investigations \\cite{Roca-Maza2017}.\nThe smearing parameter $\\epsilon$ in Eq.~(\\ref{eq:homega}) is chosen as $0.25$ MeV.\n\n\n\n\\section{Results and discussion}\\label{sec:res}\n\n\\subsection{Spectrum and sum rules}\\label{sec:str}\n\nIn Fig.~\\ref{fig:str} we show the strength function of ISGMR, ISGDR, and ISGQR in $^{16}$O calculated by RPA (bars) and PVC (lines), in comparison with experimental data \\cite{Harakeh1981,Lui2001}. The original data is given in terms of the fraction of EWSR $F(E)$ in Ref.~\\cite{Lui2001}, with a total of $(48\\pm 10)\\%$, $(32\\pm 7)\\%$, and $(53\\pm10)\\%$ of the EWSR in the region $E_x$ from 11 to 40 MeV. This data is transformed to the strength distribution by:\n\\begin{equation}\\label{eq:fe}\n S(E) = \\frac{F(E)}{E} m_1,\n\\end{equation}\nwith the values of $m_1$ adopted as the double commutator ones in Table~\\ref{tab:sr}.\nFor the dipole resonance, the level at $7.12$ MeV which exhausts $4.2\\%$ of the EWSR is taken from Ref. \\cite{Harakeh1981}.\n\nIn previous studies of PVC such as Refs.~\\cite{Colo1994,Roca-Maza2017}, the interaction vertex $Q_1HQ_2$ in Eq.~(\\ref{eq:q1hq2}) includes only the central term of the Skyrme interaction.\nThe effect of other terms on the single-particle properties have been investigated in Ref.~\\cite{Colo2010,CaoLG2014}.\nHere we would like to investigate the effect of those terms on the strength function, therefore in Fig.~\\ref{fig:str} both the results of PVC with central interaction ($V_c$) and with full interaction ($V_{\\rm full}$) are given, within the diagonal approximation (PVC-dia).\nFor PVC without diagonal approximation (PVC), only the results with full interaction are given. In all cases, the HF+RPA calculations are performed with full Skyrme interaction.\n\nIt can be seen from Fig.~\\ref{fig:str} that by including the escape and spreading effects within the PVC, the width of the strength distribution appears naturally, unlike in the case of RPA. This makes the comparison with experimental data more realistic. On the other hand, the centroid of the distribution ($m_1\/m_0$) is shifted to a lower energy, from few hundreds of keV for the ISGMR and ISGDR to a maximum of about 1.5 MeV for the case of the ISGQR (cf. Table \\ref{tab:sr}). It is important to note here that functionals are usually calibrated in order to give a reasonable description of the experimental centroid energy at the RPA level and, therefore, such shift may lead to worse agreement with the data.\n\nBy comparing the results with central term only and results with full interaction, it can be seen that by including Coulomb term and spin-orbit term, the strength is generally slightly shifted to a lower energy. In the case of ISGMR and ISGDR, the shape of the strength distribution does not change too much, while in ISGQR such change is more significant.\n\nFrom PVC-dia to PVC, the strength function is also much influenced in the ISGQR case. For PVC-dia, there are two major peaks near 17 and 18.5 MeV; while for PVC, there are four major peaks near 14.5, 17, 18.5, and 19.5 MeV, with lower strength and wider distribution. The lowest peak near 14.5 MeV is of particular interest as there is no sign of this peak in PVC-dia. It will be used as an example in Section \\ref{sec:dia} to analyze the difference between calculation with and without the diagonal approximation. Regarding the ISGMR, the removal of the diagonal approximation also shows some effect, for example: the lowest peak near 16.5 MeV is slightly shifted to a lower energy and the strength increases; The distributions of the peaks near 20, 21.5, 24, and 29.5 MeV are also affected, but, overall, the effect is weaker than the case of ISGQR. Among the three cases, the ISGDR is the one where the diagonal approximation shows less influence.\n\n\\begin{figure*}[htbp]\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16mons.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16isdips.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16quas.eps}\n \\hspace{-0.2cm}\n \\caption{(Color online) Same as Fig.~\\ref{fig:str}, but for comparison between PVC calculation with and without subtraction.}\n \\label{fig:str2}\n\\end{figure*}\n\nAs mentioned in the Introduction, the diagonal approximation has been investigated in the SRPA framework for the giant resonances of $^{16}$O in Ref.~\\cite{Gambacurta2010}.\nFrom RPA to SRPA, the strength distributions are shifted towards a lower energy, similar to the effect of PVC in Fig.~\\ref{fig:str}.\nHowever, quantitatively, the effect of SRPA is larger.\nFor ISGMR, IVGDR, and ISGQR, the main peaks are shifted towards a lower energy by about 4, 6, and 8 MeV, respectively ~\\cite{Gambacurta2010}; while for PVC the shifts are $\\approx 2-3$ MeV.\nFor the ISGMR, the diagonal approximation in SRPA shifts the distribution to a lower energy by about 2 MeV, while in PVC it changes mildly (see Fig.~\\ref{fig:str} (a) of this work and Fig. 8 (a) of Ref.~\\cite{Gambacurta2010}).\nFor the dipole case, the effect of diagonal approximation is small in both SRPA and PVC (see Fig.~\\ref{fig:str} (b) of this work and Fig. 15 (b) of Ref.~\\cite{Gambacurta2010}).\nFor the ISGQR, the diagonal approximation in SRPA shifts the distribution towards a lower energy by around 2 MeV, similar to ISGMR, while in PVC it is more complicated as the shape has changed much (see Fig.~\\ref{fig:str} (c) of this work and Fig. 9 (a) of Ref.~\\cite{Gambacurta2010}).\nIn all the cases, the diagonal approximation in SRPA does not change much the shape of the strength distribution, while in PVC this is not the case for the ISGQR.\n\nThe diagonal approximation has also been studied in the RTBA framework \\cite{Litvinova2010,Litvinova2013}, where it was removed by including additional phonon coupling between two quasiparticles inside the 2 quasiparticles $\\otimes$ phonon configuration.\nThe low-lying dipole excitations in $^{116,120}$Sn and $^{68,70,72}$Ni were investigated.\nBy removing the diagonal approximation, a larger fraction of the pygmy mode is pushed above the neutron threshold.\nFor a detailed comparison between RTBA and current framework, see Appendix \\ref{app:tba}.\n\nIn Fig.~\\ref{fig:str2} the effects of the subtraction [Eq.~(\\ref{eq:sub})] in the PVC calculation are shown for the ISGMR, ISGDR, and ISGQR strength distributions of $^{16}$O. As a reference, the results of the RPA and experimental data shown in Fig.~\\ref{fig:str} are also displayed in Fig.~\\ref{fig:str2}. It can be seen that by adopting the subtraction procedure, the strength distributions are generally shifted towards a higher energy by about 1 MeV, except in the ISGDR case where the main peak at 17 MeV vanishes and a new peak at 14 MeV appears.\nThe effects of subtraction presented here are consistent with the findings of previous investigation using PVC-dia (see Fig. 4 of Ref.~\\cite{Roca-Maza2017}).\n\nIn Ref.~\\cite{Lyutorovich2015}, the quasiparticle-phonon coupling model with time-blocking approximation was used to study the ISGMR, ISGQR, and isovector GDR of $^{16}$O, $^{40}$Ca, and $^{208}$Pb.\nA systematic downward shift of the centroid energy of the giant resonances was found from RPA to TBA with subtraction.\nThis effect is similar to the one of PVC presented here, though quantitatively it is smaller (see Fig.~\\ref{fig:str2} of this work and Fig. 4 of Ref.~\\cite{Lyutorovich2015}). Especially in the case of ISGQR, the PVC calculation (with or without subtraction) gives very different strength distribution from the one given by RPA, while they are similar for TBA and RPA \\cite{Lyutorovich2015}.\nThis might be related with the diagonal approximation as removing it shows quite some effect here.\n\nThe subtraction in SRPA has been investigated for ISGMR and ISGQR and it also pushes the strength distribution to a higher energy \\cite{Gambacurta2015}.\nHowever, comparing with the results of PVC shown in Fig.~\\ref{fig:str2}, the effect in SRPA is again larger.\nWith subtraction, the strength are shifted towards a higher energy by about 2 MeV in SRPA while in PVC it is generally less than 1 MeV, see Figs. 1 and 4 of Ref.~\\cite{Gambacurta2015}.\nComparing the results of SRPA including subtraction with RPA given in Ref.~\\cite{Gambacurta2015}, the main peaks of ISGMR and ISGQR given by SRPA with subtraction are about 1.5 and 1 MeV lower than those by RPA.\nThese are similar to the differences between PVC with subtraction and RPA shown in Fig.~\\ref{fig:str2}.\n\nIn comparison with the experimental data, the three peaks of ISGMR around 18, 23, and 26 MeV may correspond to the three peaks given by the PVC, though the energies are slightly lower than the data.\nThis may be understood as the SAMi functional has been developed in such a way that the experimental ISGMR is reproduced at the RPA level, which can be seen from the black vertical lines in Fig.~\\ref{fig:str2}.\nWhen the PVC is included, though the description of resonance width has been improved, the centroid is pushed to a slightly lower energy and the subtraction remedies to this problem only to some extent.\nThe peaks around 12 and 14 MeV may be due to $\\alpha$-clustering effects \\cite{Yamada2012}.\n\nIn the case of ISGDR the description of PVC with SAMi functional is rather good, especially the low-lying 7.1 MeV level has been nicely reproduced.\nThe peaks around 12 and 18 MeV, and the resonance shape above 20 MeV are also well described.\nThe subtraction worsens the description of the data below 20 MeV.\n\nIn the case of the ISGQR, the peak at 15 MeV by PVC might be attributed to the peak at 12 MeV or 14 MeV of the data.\nThe experimental data for the high energy part of ISGQR is concentrated from 18 to 26 MeV, while the theoretical distribution is from 16 to 22 MeV, slightly lower than the data. Again, the strength distribution given by subtraction method is shifted towards higher energy, but the peak position is still lower than the experimental data.\nIt has been shown that the dominant decay channel of ISGQR of $^{16}$O is $\\alpha$ emission to the ground and first excited states of $^{12}$C \\cite{Knopfle1978}.\nSuch effect is not included in current PVC framework and could be part of the reasons for the disagreement with the data.\nBesides that, taking into account the coupling with more phonons \\cite{Litvinova2018a} and ground-state correlation might also help to improve the descriptions of the ISGMR and ISGQR of $^{16}$O.\n\nThe sum rules for the above discussed calculations are shown in Table \\ref{tab:sr}, including the results for the RPA, PVC-dia without Coulomb and spin-orbit interactions in the PVC vertex (PVC-d, $V_c$), PVC-dia with full interaction (PVC-d), PVC, and PVC with subtraction (PVC-s).\nThe strength function given by PVC are integrated up to $E = 120$ MeV.\nThe EWSR $m_1$ by the double commutator in Eq.~(\\ref{eq:m1dc}) are also shown.\n\n\\begin{table}[h]\n \\caption{Sum rules for the ISGMR, ISGDR and ISGQR responses in $^{16}$O calculated by: RPA, PVC-dia without Coulomb and spin-orbit interactions (PVC-d, $V_c$), PVC-dia with full interaction (PVC-d), PVC, PVC with subtraction (PVC-s).\n The EWSR $m_1$ by double commutator (DC) is also given.\n In all cases the SAMi functional is used.\n The units of $m_{-1}, m_0$, and $m_1$ are fm$^4$\/MeV, fm$^4$, and fm$^4$ MeV, respectively, for the ISGMR and ISGQR; they are \n fm$^6$\/MeV, fm$^6$, and fm$^6$ MeV for the ISGDR.\n }\n \\label{tab:sr}\n \\centering\n \\begin{ruledtabular}\n \\begin{tabular}{l|lcccccc}\n & SR & RPA & DC & PVC-d,$V_c$ & PVC-d & PVC & PVC-s \\\\\n \\hline\n ISGMR & $m_{-1}$ & 1.14 & & 1.25 & 1.25 & 1.25 & 1.17 \\\\\n & $m_0$ & 27.3 & & 27.9 & 27.8 & 27.8 & 27.8 \\\\\n & $m_1$ & 689 & 688 & 701 & 701 & 700 & 740 \\\\\n & $m_{1}\/m_0$ & 25.3 & & 25.2 & 25.2 & 25.2 & 26.6 \\\\\n&$\\sqrt{\\frac{m_{1}}{m_{-1}}}$& 24.6 & & 23.7 & 23.7 & 23.7 & 25.1 \\\\\n \\hline\n ISGDR & $m_{-1}$ & 38.4 & & 42.5 & 43.0 & 43.0 & 42.8 \\\\\n & $m_0$ & 968 & & 981 & 982 & 981 & 1008 \\\\\n & $m_1$ & 29567 &29493& 29583& 29619 & 29591 & 30607 \\\\\n & $m_{1}\/m_0$ & 30.5 & & 30.2 & 30.2 & 30.2 & 30.4 \\\\\n&$\\sqrt{\\frac{m_{1}}{m_{-1}}}$& 27.7 & & 26.4 & 26.2 & 26.2 & 26.7 \\\\\n \\hline \n ISGQR & $m_{-1}$ & 18.4 & & 22.6 & 23.2 & 23.3 & 20.9 \\\\\n & $m_0$ & 397 & & 419 & 420 & 420 & 413 \\\\\n & $m_1$ & 8613 &8604& 8503 & 8488 & 8489 & 8949 \\\\\n & $m_{1}\/m_0$ & 21.7 & & 20.3 & 20.2 & 20.2 & 21.7 \\\\\n&$\\sqrt{\\frac{m_{1}}{m_{-1}}}$& 21.6 & & 19.4 & 19.1 & 19.1 & 20.7 \\\\\n \\end{tabular}\n \\end{ruledtabular}\n\\end{table}\nFirst, the EWSR ($m_1$) values given by the RPA calculation in all three cases, the ISGMR, ISGDR, and ISGQR, are fully exhausted comparing with the ones obtained by the double commutator.\nFor the PVC results, there are small discrepancies.\nAs discussed in Sec. \\ref{sec:sr}, when only spreading term is taken into account, the $m_0$ and $m_1$ given by PVC should be the same as those of RPA, similar to the case of SRPA \\cite{Adachi1988}.\nTo verify this, we show in Fig.~\\ref{fig:sr} the $m_0$ and $m_1$ as a function of the upper limit of the integrated energy range.\nThe case of ISGMR is taken as an example, while the others give similar results.\nIn this figure the horizontal dashed line is the RPA results for $m_0$ and DC for $m_1$ (see Table \\ref{tab:sr}).\nOne can see that when only spreading term ($W^\\downarrow$) is included in the PVC calculation, the $m_0$ and $m_1$ are the same as those of RPA.\n\nWhen escape term ($W^\\uparrow$) is included, the sum rules $m_0$ and $m_1$ are slightly different from those of RPA.\nThis could be due to the approximation in the escape term that the interaction has not been taken into account \\cite{Colo1994}.\nTo verify this, we compare the sum rules of PVC with escape term only starting from RPA phonons, to the results that starting from unperturbed phonons (without particle-hole interactions).\nThe latter one is labeled UNP and the results are listed in Table \\ref{tab:sr2}.\nTo achieve a higher precision, the smearing parameter $\\epsilon$ in Eq.~(\\ref{eq:homega}) is chosen as $0.1$ MeV in this Table instead of 0.25 MeV in other calculations.\nIt can be seen that when interactions are not taken into account at the beginning, the $m_0$ and $m_1$ given by PVC with escape term are almost the same as original calculation in $Q_1$ subspace, with a relative difference 0.2\\%.\nWhen interaction is included in $Q_1$ subspace (RPA calculation) but not in $P$ subspace (PVC with escape term), the sum rules will slightly be influenced.\n\n\\begin{table}[h]\n \\caption{Sum rules for the ISGMR in $^{16}$O calculated by RPA (or unperturbed calculation, UNP) and PVC with escape term only $(W^\\uparrow)$.\n The relative differences between PVC and RPA (or UNP) are also given $(\\delta)$.\n In all cases the SAMi functional is used.\n }\n \\label{tab:sr2}\n \\centering\n \\begin{ruledtabular}\n \\begin{tabular}{l|ccc|ccc}\n SR & RPA & $W^\\uparrow$ & $\\delta$ & UNP & $W^\\uparrow$ & $\\delta$ \\\\\n \\hline\n $m_0$ (fm$^4$) & 27.27 & 27.88 & 2.2\\% & 28.13 & 28.06 & 0.2\\% \\\\\n $m_1$ (fm$^4$MeV) & 689.1 & 702.8 & 2.0\\% & 816.8 & 815.5 & 0.2\\% \\\\\n \\end{tabular}\n \\end{ruledtabular}\n\\end{table}\n\nAs it has been discussed in Fig.~\\ref{fig:str}, the strength distributions given by PVC are generally shifted to a lower energy comparing with those by RPA . Therefore, the inverse EWSR ($m_{-1}$) are larger compared with RPA even when the subtraction method is implemented, and the centroid energy ($m_1\/m_0$ or $\\sqrt{m_1\/m_{-1}}$) are smaller. The influence of non-central terms of the interaction (comparing PVC-d,$V_c$ and PVC-d) on the sum rules are negligible in the case of ISGMR, while for ISGDR and ISGQR a small effect shows up.\nIn all cases, the diagonal approximation (comparing PVC-d and PVC) has little influence on the sum rules.\nOn the other hand, the subtraction has much influence on the sum rules (comparing PVC and PVC-s).\nThe EWSR are significantly larger and agree less with the double commutator sum rule when subtraction is performed, in agreement with the findings in the PVC-dia calculation \\cite{Roca-Maza2017} and SRPA \\cite{Gambacurta2015}. This is a feature of the subtraction method that needs to be better investigated. We recall here that the subtraction method was devised for exactly keeping the $m_{-1}$ value obtained within the RPA in beyond RPA calculations while no procedure of renormalization was imposed on $m_1$.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{sr-o16mon.eps}\n \\caption{(Color online) Sum rules $m_0$ and $m_1$ of ISGMR in $^{16}$O as a function of the upper limit of the integrated energy range, calculated by PVC with escape term ($W^\\uparrow$), spreading term ($W^\\downarrow$), and both terms.\n Horizontal black dashed lines are RPA results (for $m_0$) and DC results (for $m_1$), see Table \\ref{tab:sr} for the values.}\n \\label{fig:sr}\n\\end{figure}\n\n\n\\subsection{Different components of interaction}\n\nNext, we show how different components of the interaction contribute to the strength function in Fig.~\\ref{fig:str}, using the ISGQR as an example.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{o16qua-vsocou.eps}\n \\caption{(Color online) Strength function of ISGQR in $^{16}$O calculated by RPA with full interaction, PVC with central interaction (PVC, $V_c$), PVC with central plus Coulomb interactions (PVC, $V_c + V_{\\rm Cou.}$), PVC with central plus spin-orbit interactions (PVC, $V_c + V_{\\rm s.o.}$), and PVC with full interaction (PVC, $V_{\\rm full}$).\n In all cases the SAMi functional is used.}\n \\label{fig:strvso}\n\\end{figure}\n\nIn Fig.~\\ref{fig:strvso} the strength distributions calculated by PVC and with different terms of interaction are shown, including: with central terms only ($V_c$), with central terms and Coulomb term ($V_{c}+V_{\\rm Cou.}$), with central terms and spin-orbit term ($V_c+V_{\\rm s.o.}$), and with full interaction ($V_{\\rm full}$).\nIt can be seen that the Coulomb interaction has a negligible effect on the strength distribution, except for a small influence near 14 and 21 MeV.\nOn the other hand, the spin-orbit term has much influence and clearly changes the distribution.\nWith the central term there is only one minor peak near 14 MeV and one major peak near 18.5 MeV.\nWhen including the spin-orbit term, the strength of the major peak decreases much and two other peaks near 17 and 19.5 MeV become larger.\n\n\n\\subsection{Influence of diagonal approximation: eigen-energies}\\label{sec:dia}\n\nIn this Subsection, we analyze the difference between PVC with and without diagonal approximation in Fig.~\\ref{fig:str}.\nThe low energy peak at $\\omega = 14.6$ MeV in the ISGQR will be used as an example, as it \nis manifestly different in the two calculations.\nIn the following results, all PVC calculations are performed with the full interaction at $\\omega = 14.6$ MeV.\nThe integration of the strength around this energy ($14.6 \\pm 0.4$ MeV) within PVC calculation gives \n$\\int S_{\\rm PVC} d\\omega = 34.9$ fm$^4$, whereas within PVC-dia is $\\int S_{\\text{PVC-dia}} d\\omega = 3.7$ fm$^4$.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{str-q1460.eps}\n \\caption{(Color online) Contribution to the ISGQR strength function of $^{16}$O at $\\omega = 14.6$ MeV from different PVC eigenstates $\\nu$ with different excitation energies, see also Eq.~(\\ref{eq:str}).\n Results of (a) PVC (red lines) and (b) PVC-dia (blue lines) are shown.\n The position of $\\omega = 14.6$ MeV is given by the vertical dashed line, and the RPA states are given by the gray vertical lines (with unit fm$^4$).}\n \\label{fig:strn}\n\\end{figure}\n\nFigure \\ref{fig:strn} shows the strength contributions from different PVC states $\\nu$, as given by Eq.~(\\ref{eq:str}).\nThe position of $\\omega = 14.6$ MeV has been indicated by the vertical dashed line, and the RPA states are given by the gray vertical lines in the background.\nFor PVC, the largest contribution to the strength at $\\omega = 14.6$ MeV comes from the PVC state at $\\Omega_\\nu = 15.5$ MeV; while for PVC-dia, the largest contribution comes from the state at $\\Omega_\\nu = 18.0$ MeV.\n\nLet us express the square of the transition matrix elements explicitly in terms of its real and imaginary parts,\n\\begin{equation}\\label{eq:}\n \\langle 0|O|\\nu \\rangle^2 = a_\\nu + ib_\\nu,\n\\end{equation}\nwith $a_\\nu$ and $b_\\nu$ both real numbers.\nFrom Eq.~(\\ref{eq:str}), the strength function can be written as\n\\begin{equation}\\label{eq:str2}\n S(\\omega) = \\frac{1}{\\pi} \\sum_\\nu \\frac{a_\\nu\\frac{\\Gamma_\\nu}{2}-b_\\nu(\\omega-\\Omega_\\nu)}{(\\omega-\\Omega_\\nu)^2+\\frac{\\Gamma_\\nu^2}{4}}.\n\\end{equation}\nIn Fig.~\\ref{fig:o0n} \nthe real part of the square of the transition matrix element $a_\\nu =$ Re($\\langle 0|O|\\nu \\rangle^2$) is shown.\nIt can be seen that both the transition matrix element \nof the state at $\\Omega_\\nu = 15.5$ MeV in PVC, \nand that of the state at $\\Omega_\\nu = 18.0$ MeV in PVC-dia, are very large.\nAlthough the value of the $\\Omega_\\nu({\\rm PVC}) = 15.5$ MeV one is slightly larger than the one of the $\\Omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV, the difference is not large enough to explain the difference in the final contribution to the strength shown in Fig.~\\ref{fig:strn}.\nTherefore, according to Eq.~(\\ref{eq:str2}), the much stronger strength in the PVC from $\\Omega_\\nu({\\rm PVC}) = 15.5$ MeV state must be due to the position of this state, which is much closer to the energy being evaluated, that is, $\\omega = 14.6$ MeV.\nIn this way, the energy denominator in Eq.~\\ref{eq:str2} of this state is much smaller than the $\\Omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV state and as a consequence the strength is larger.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{o0n-q1460.eps}\n \\caption{(Color online) Real part of the square of the transition matrix element $\\langle 0|O|\\nu \\rangle$ for PVC states $\\nu$ with different excitation energies.\n The calculation is performed for the ISGQR of $^{16}$O at $\\omega = 14.6$ MeV (vertical dashed line) by (a) PVC and (b) PVC-dia.}\n \\label{fig:o0n}\n\\end{figure}\n\nNext, we will study the origin of the large difference in the eigenenergies of these two states.\nFirst, one needs to identify the components of these two states, or more specifically, from which RPA states they come from.\nFor this purpose, we will identify them by looking at the corresponding wave functions.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{qwfon1460.eps}\n \\caption{(Color online) (a) Real part of the PVC wave function (\\ref{eq:wf}) of states $\\Omega_\\nu({\\rm PVC}) = 15.5$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV in the basis of RPA states $|n\\rangle$ by PVC and PVC-dia.\n (b) Transition matrix elements $\\langle n|O|0 \\rangle$.}\n \\label{fig:wf}\n\\end{figure}\n\nIn the upper panel of Fig.~\\ref{fig:wf} we show the real part of the PVC wave function $F_n^{(\\nu)}$ (\\ref{eq:wf}) of states $\\Omega_\\nu({\\rm PVC}) = 15.5$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV in the basis of RPA states $|n\\rangle$ by PVC and PVC-dia.\nThe transition matrix elements $\\langle n|O|0 \\rangle$ in the RPA representation are shown in the lower panel.\nThe transition matrix elements $\\langle 0|O|\\nu \\rangle$ in the PVC representation in Fig.~\\ref{fig:o0n} can be calculated as\n\\begin{equation}\\label{eq:0on}\n \\langle 0|O|\\nu \\rangle = \\sum_n F_n^{(\\nu)} \\langle 0|O|n \\rangle.\n\\end{equation}\n\nIt can be seen that these two states have similar RPA components in both PVC calculations.\nAlthough not really dominant, the major component of these two states can be identified as the 11th RPA state with the largest transition matrix element $\\langle n=11|O|0 \\rangle$.\nThis RPA state is the one located at $\\omega_n = 21.3$ MeV with the largest strength as shown in Fig.~\\ref{fig:str} (c) or Fig.~\\ref{fig:strn}.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{qwfph1460.eps}\n \\caption{(Color online) (a) RPA wave function ($X$ amplitude) of state with excitation energy $\\omega_n = 21.3$ MeV in the 1p-1h representation, see Eq.~(\\ref{eq:qn}).\n Different regions divided by vertical lines are for different hole states.\n (b) Real part of the PVC wave functions of states with excitation energies $\\Omega_\\nu({\\rm PVC}) = 15.5$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV, see Eq.~(\\ref{eq:fph}), by PVC and PVC-dia.}\n \\label{fig:wfph}\n\\end{figure}\n\nIn Fig.~\\ref{fig:wfph}, the wave function $F_n^{(\\nu)}$ of states $\\omega_\\nu({\\rm PVC}) = 15.5$ MeV and $\\omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV are transformed to the 1p-1h basis for the $X$ amplitude as\n\\begin{equation}\\label{eq:fph}\n F_{ph}^{(\\nu)} = \\sum_n X_{ph}^{(n)} F_n^{(\\nu)}.\n\\end{equation}\nSimilar transformation can be done for the $Y$ amplitudes, but as their values are very small they will not be shown here.\nFrom this figure it can be seen that the original RPA state $\\omega_n = 21.3$ MeV is a very collective state with many 1p-1h components involved in.\nWhen considering the escape and spreading effect of the PVC, we noticed that this state becomes even more collective.\nIn both PVC and PVC-dia calculations, the wave functions $F_{ph}^{(\\nu)}$ of these states are similar.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{wn.eps}\n \\caption{(Color online) Excitation energies of ISGQR in $^{16}$O calculated with SAMi functional by RPA, PVC, and PVC-dia.\n The diagonal PVC matrix elements before diagonalizing the PVC Hamiltonian $\\mathcal{H}_{nn}$ in Eq.~(\\ref{eq:hnn}) are also given.}\n \\label{fig:wn}\n\\end{figure}\n\nAfter identifying the major RPA components of the PVC states, one can see how the eigenenergies change from RPA to PVC.\nIn Fig.~\\ref{fig:wn} we show the RPA solutions $\\omega_n$ for the ISGQR of $^{16}$O, \nbelow 23 MeV, obtained by using the SAMi functional, together with the corresponding eigenenergies of \nthe PVC solutions at $\\omega = 14.6$ MeV.\nThe diagonal matrix elements of the PVC Hamiltonian (\\ref{eq:hnn}) before diagonalizing, $\\mathcal{H}_{nn}(\\omega) = \\omega_n + W_{nn}(\\omega)$ have also been shown, with $W = W^\\uparrow + W^\\downarrow$ the escape term plus spreading term.\nThe corresponding levels are connected with dotted lines, with bold dashed lines emphasising the link between RPA state $\\omega_n = 21.3$ MeV, and PVC states $\\Omega_\\nu({\\rm PVC}) = 15.5$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 18.0$ MeV.\n\nIt can be seen from Fig.~\\ref{fig:wn} that the diagonal PVC matrix elements are attractive.\nIn PVC-dia $W_{nn} = -2.0$ MeV while for PVC the value is $-2.7$ MeV, that is, $0.7$ MeV more attraction by removing the diagonal approximation.\nAfter diagonalizing the PVC Hamiltonian $\\mathcal{H}$, the energy level changes from the \nperturbative approximation $\\mathcal{H}_{nn}$ (originated from the $\\omega_n = 21.3$ MeV RPA state) to the final eigenvalue $\\Omega_\\nu$ with \na further decrease of \n$1.3$ MeV in PVC-dia, and of $3.1$ MeV in PVC.\nIn the end, the eigenenergy of this state in PVC-dia is $\\Omega_\\nu(\\text{PVC-dia}) = 21.3 - 2.0 - 1.3 = 18.0$ MeV, while in PVC is $\\Omega_\\nu({\\rm PVC}) = 21.3 - 2.7 - 3.1 = 15.5$ MeV.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{me.eps}\n \\caption{(Color online) PVC matrix elements $W = W^\\uparrow + W^\\downarrow$ with index $n$ referring to the RPA basis.\n Results are shown for PVC and PVC-dia for the ISGQR of $^{16}$O at excitation energy $\\omega = 14.6$ MeV.\n The 11th RPA state is the one with excitation energy $\\omega_{n=11} = 21.3$ MeV and being discussed in the text.}\n \\label{fig:me}\n\\end{figure}\n\nIn Fig.~\\ref{fig:me}, the PVC matrix elements $W$ are shown with the index $n$ referring to the RPA basis.\nSince the numbering for the RPA state we are interested in is $n = 11$, with excitation energy $\\omega_n = 21.3$ MeV, the matrix elements are shown for $W_{n,11}$.\nIn this figure, the big attraction of the diagonal matrix elements $W_{11,11}$ for both calculations can be clearly seen, with $0.7$ MeV more in PVC calculation.\nMoreover, the magnitudes of the nondiagonal matrix elements are generally larger PVC calculation, which in the end leads to more mixing of other states and lower eigenvalues after diagonalizing.\n\nAs a conclusion, the extra attraction shown by removing the diagonal approximation in the PVC model is the main cause of the appearance of the low energy peak in the ISGQR. \n\n\n\\subsection{Influence of diagonal approximation: coupling between neutron and proton particle-hole configurations}\\label{sec:dia2}\n\nIn this subsection, we analyze another important difference between PVC with and without diagonal approximation shown in Fig.~\\ref{fig:str}, that is, the coupling of neutron 1p-1h excitations and proton 1p-1h excitations.\nThe low energy peak at $\\omega = 16.4$ MeV in the ISGMR will be used as an example.\nThe integration of the strength around this energy ($16.4 \\pm 0.4$ MeV) by PVC is $\\int S_{\\rm PVC} d\\omega = 1.5$ fm$^4$, and by PVC-dia is $\\int S_{\\text{PVC-dia}} d\\omega = 1.3$ fm$^4$.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{str-m1640.eps}\n \\caption{(Color online) Similar as Fig.~\\ref{fig:strn}, but showing the contributions to the ISGMR strength function of $^{16}$O at $\\omega = 16.4$ MeV from different PVC eigenstates $\\nu$ with different excitation energies.\n Results of (a) PVC (red lines) and (b) PVC-dia (blue lines) are shown.\n The position of $\\omega = 16.4$ MeV is given by the vertical dashed line, and the contributions from RPA states are given by the gray vertical lines (with unit fm$^4$).}\n \\label{fig:strn2}\n\\end{figure}\n\nFigure \\ref{fig:strn2} shows the strength contributions from different PVC states $\\nu$, as given by Eq.~(\\ref{eq:str}).\nThe position of $\\omega = 16.4$ MeV has been indicated by the vertical dashed line, and the \ncontributions from the RPA states are given by the gray vertical lines in the background.\nFor PVC, the largest contribution to the strength at $\\omega = 16.4$ MeV comes from the state at \n$\\Omega_\\nu = 16.7$ MeV while, while for PVC-dia, the largest contribution comes from the state at \n$\\Omega_\\nu = 17.2$ MeV.\nAt variance with the situation discussed in Fig.~\\ref{fig:strn}, these two PVC states are both close to the energy being evaluated ($\\omega = 16.4$ MeV).\nTherefore, from Eq.~(\\ref{eq:str}), one can hint that the difference in the strength should come from the difference in the transition matrix element in these two calculations.\n\nIn Fig.~\\ref{fig:o0n2} the real part of the square of the transition matrix element Re($\\langle 0|O|\\nu \\rangle^2$) is shown.\nAs expected, the transition matrix element of the state $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV is larger than the state $\\Omega_\\nu(\\text{PVC-dia}) = 17.2$ MeV, and this explains the larger strength in PVC.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{o0n-m1640.eps}\n \\caption{(Color online) Real part of the square of the transition matrix element $\\langle 0|O|\\nu \\rangle$ for PVC states $\\nu$ with different excitation energies.\n The calculation is performed for the ISGMR of $^{16}$O at $\\omega = 16.7$ MeV (vertical dashed line) by (a) PVC and (b) PVC-dia.}\n \\label{fig:o0n2}\n\\end{figure}\n\nTo understand the difference in the transition matrix elements, we show in Fig.~\\ref{fig:wf2} the PVC wave function $F_n^{(\\nu)}$ (\\ref{eq:wf}) of states $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 17.2$ MeV in the basis of RPA phonons $|n\\rangle$, and the transition matrix elements $\\langle n|O|0 \\rangle$ in the RPA representation.\nThe transition matrix elements $\\langle 0|O|\\nu \\rangle$ in the PVC representation in Fig.~\\ref{fig:o0n2} can be calculated \nas in Eq.~(\\ref{eq:0on}).\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{mwfon1640.eps}\n \\caption{(Color online) (a) Real part of the PVC wave function (\\ref{eq:wf}) of states $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 17.2$ MeV in the basis of RPA states $|n\\rangle$ by PVC and PVC-dia.\n (b) Transition matrix elements $\\langle n|O|0 \\rangle$.}\n \\label{fig:wf2}\n\\end{figure}\n\nDifferently, again, from the situation of the lowest peak in the ISGQR, these two states in Fig.~\\ref{fig:wf2} have very different RPA components.\nIn PVC-dia, the major RPA component of the state $\\Omega_\\nu(\\text{PVC-dia}) = 17.2$ MeV can be identified as the 4th RPA phonon, while the state $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV has the same major component but very much mixed with the 6th RPA phonon.\nSince the 6th RPA phonon has a larger transition matrix element than the 4th RPA phonon, according to Eq.~(\\ref{eq:0on}), the transition matrix element for $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV is also larger.\n\nIn Fig.~\\ref{fig:wfph2} we show the 1p-1h components of these two PVC states as well as the related RPA states $\\omega_{n=4} = 18.6$ MeV and $\\omega_{n=6} = 19.8$ MeV.\nAs mentioned above, the PVC state $\\Omega_\\nu(\\text{PVC-dia}) =17.2$ MeV is dominated by the RPA phonon $\\omega_{n} = 18.6$ MeV and therefore its wave function in the 1p-1h representation is very similar to this phonon.\nFor PVC state $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV, the RPA component $\\omega_{n} = 18.6$ MeV, which is mainly a neutron $p_{1\/2}$ excitation, is very much mixed with the component $\\omega_{n} = 19.8$ MeV, which is mainly a proton $p_{1\/2}$ excitation. \nIn other words, in PVC there is a coupling between a neutron 1p-1h excitation and a proton 1p-1h excitation, which can not show up in PVC-dia as we discuss in detail in what follows.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{mwfph1640.eps}\n \\caption{(Color online) RPA wave function ($X$ amplitudes) of the state with excitation energy (a) $\\omega_n = 18.6$ MeV and (b) $\\omega_n = 19.8$ MeV in the 1p-1h representation, see Eq.~(\\ref{eq:qn}).\n Different regions divided by vertical lines are for different hole states.\n (c) Real part of the PVC wave functions of states with excitation energies $\\Omega_\\nu({\\rm PVC}) = 16.7$ MeV and $\\Omega_\\nu(\\text{PVC-dia}) = 17.2$ MeV, see Eq.~(\\ref{eq:fph}), by PVC and PVC-dia.}\n \\label{fig:wfph2}\n\\end{figure}\n\nThe reason for the coupling between neutron and proton 1p-1h states is shown in Fig.~\\ref{fig:wdph3d}, in which the matrix elements of the spreading term in PVC and PVC-dia are plotted.\nThe index $ph$ is the same as in Fig.~\\ref{fig:wfph2}.\nAs expected, most of the matrix elements are attractive (negative values) and therefore the strength distributions are shifted to a lower energy.\n\n\\begin{figure}[htbp]\n \\includegraphics[width=8cm]{wdph1.eps}\n \\includegraphics[width=8cm]{wdph2.eps}\n \\caption{(Color online) Matrix elements of spreading term in the 1p-1h representation in (a) PVC (\\ref{eq:wph}) and (b) PVC-dia (\\ref{eq:wph2}).}\n \\label{fig:wdph3d}\n\\end{figure}\n\nThe general pattern of the matrix is similar for both PVC and PVC-dia.\nHowever, there is a clear difference that the matrix elements of neutron-proton interaction in PVC are nonzero while in PVC-dia they are zero.\nThis can be understood from the expression of spreading term in Eq.~(\\ref{eq:wph}) and the diagram in Fig.~\\ref{fig:q1hq2}.\nThe matrix element $A_{p'h',p_1'h_1'n}$ in Eq.~(\\ref{eq:wph}) (or $Q_1HQ_2$ in Fig.~\\ref{fig:q1hq2}) can not couple the initial 1p-1h excitation ($p_1'h_1'$) with the final 1p-1h excitation ($p'h'$) that has a different charge, and the same is true for $A_{p_1h_1n,ph}$.\nOnly in the denominator $A_{p_1'h_1'n,p_1h_1n}$ (or $Q_2HQ_2$ in Fig.~\\ref{fig:q1hq2}) there is interaction $(\\bar{V}_{p_1h_2h_1p_2})$ between the initial and final 1p-1h excitations with different charges.\nWhen the diagonal approximation is applied in the denominator $Q_2HQ_2$, this interaction is removed and as a consequence the spreading term has zero matrix elements in the off-diagonal blocks where the neutron and proton 1p-1h excitations interact.\n\nIn the case of ISGQR discussed in Fig.~\\ref{fig:wfph}, the original RPA phonon is already composed of many neutron and proton 1p-1h excitations. Therefore in that case the coupling between 1p-1h states of different charges nature via the denominator in Eq.~(\\ref{eq:wph}) is not significant.\n\n\n\\subsection{Dependence of different functionals}\n\n\\begin{figure*}[htbp!]\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16mon-ski3.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16dipis-ski3.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16qua-ski3.eps} \\\\\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16mon-ski3s.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16dipis-ski3s.eps}\n \\hspace{-0.5cm}\n \\includegraphics[width=6.3cm]{str-o16qua-ski3s.eps}\n \\caption{(Color online) Strength function of ISGMR (a,d), ISGDR (b,e), and ISGQR (c,f) in $^{16}$O calculated by PVC without (a-c) and with (d-f) subtraction using different Skyrme functionals.\n See Fig.~\\ref{fig:str} for the detail of the experimental data \\cite{Harakeh1981,Lui2001}.}\n \\label{fig:str3}\n\\end{figure*}\n\nTo test the dependence of the results on the choice of different functionals, in Fig.~\\ref{fig:str3} we show the strength function of ISGMR, ISGDR, and ISGQR in $^{16}$O calculated by PVC with and without subtraction using different Skyrme functionals: SAMi \\cite{Roca-Maza2012}, SIV \\cite{Beiner1975}, SkI3 \\cite{Reinhard1995}, and SKX \\cite{Brown1998}.\nAs it can be seen from the figure, the basic features such as the shape of the strength distributions obtained with different functionals are similar to each other, while in detail the results depend very much on the selected functional.\nTaking the lowest peak in (a-c) of Fig.~\\ref{fig:str3} as an example, SKX gives the lowest energy in all three cases: around 12 MeV in ISGMR, 3.5 MeV in ISGDR, and 12.5 MeV in ISGQR;\nSIV gives the highest energy, around 18 MeV in ISGMR, 9 MeV in ISGDR, and 15.5 MeV in ISGQR;\nSAMi and SkI3 sit in between with SkI3 gives slightly lower energy.\n\nFor ISGMR, if the states near 12 and 14 MeV are attributed to cluster vibrations \\cite{Yamada2012}, the rest of the resonance strength around 18, 24, and 31 MeV is best described by the SIV functional.\nFor ISGDR, SIV gives a very strong lowest excitation near 9 MeV, in agreement with the strong experimental excitation near 7 MeV; however, the strength from 12 MeV to 24 MeV by SIV is not described as well as with the other functionals.\nThe dependence on the functional in the case of the ISGDR when $E > 30$ MeV is very small.\nIn the case of the ISGQR, SIV and SkI3 give a better description, from the excitation near 15 MeV, to 19, 21, and 25 MeV.\nThe strong strength near 12 MeV given by SKX is in agreement with the data, but this model gives a too large strength near 17 MeV where no experimental peak appears.\n\nWhen subtraction is included in the PVC calculations shown in (d-f) of Fig.~\\ref{fig:str3}, the effect for different functionals are similar to the one that has been investigated in Sec. \\ref{sec:str} using SAMi.\nThe strength distributions are generally shifted to a higher energy by about $\\approx 1-2$ MeV.\nFor ISGMR, the first two peaks' positions given by SIV are now slightly higher than the data, while the major peak near $\\approx 22-24$ MeV given by SkI3 is in good agreement with the data.\nFor the ISGDR, the strength distribution given by SKX has been improved, and it describes well the experimental structures near \n7, 11, and $19-22$ MeV.\nIn this case the results from other functionals are not as good as they were before subtraction.\nFor the ISGQR, the description by SkI3 is improved with subtraction and the peaks near 19 and 21 MeV are in good agreement with the data.\n\nThe correlation between the excitation energy calculated at RPA level and nuclear matter properties has been extensively studied (cf. \\cite{Roca-Maza2018a} and references therein).\nFor instance, the compression mode ISGMR and ISGDR are correlated with the incompressibility coefficient $K_\\infty$; the ISGQR is correlated with the effective mass $m^*\/m$.\nSuch correlations still persist for the excitation energies calculated by PVC.\nFor example, among the four functionals, SAMi gives the smallest incompressibility coefficient with $K_\\infty^{\\rm (SAMi)} = 245$ MeV while SIV gives the largest $K_\\infty^{\\rm (SIV)} = 325$ MeV.\nAccordingly, the constrained energy $E_c = \\sqrt{m_1\/m_{-1}}$ given by these two functionals are $E_c^{\\rm (SAMi)} = 23.7$ MeV and $E_c^{\\rm (SIV)} = 26.7$ MeV for ISGMR, $E_c^{\\rm (SAMi)} = 26.2$ MeV and $E_c^{\\rm (SIV)} = 28.5$ MeV for ISGDR.\nSKX gives the largest effective mass with $m_{\\rm SKX}^*\/m = 0.99$ and SIV the smallest $m_{\\rm SIV}^*\/m = 0.47$.\nAccordingly, the centroid energy of ISGQR given by SKX is 15.0 MeV while by SIV is 21.6 MeV.\nThese relations are also reflected in Fig.~\\ref{fig:str3}.\n\nIn short, first, for current PVC calculation it is difficult to find a functional which can give a satisfactory description for all the giant resonances studied here.\nSecond, for different Skyrme functionals, the correlation between the excitation energy calculated by RPA and nuclear matter properties still holds for the excitation energy calculated by PVC.\n\n\n\\section{Summary}\\label{sec:sum}\n\nIn this work we have developed the self-consistent particle-vibration model without the diagonal approximation.\nThe interaction between the two particle-holes inside the doorway states has been taken into account, and it is shown that it can be derived from the equation-of-motion method similar to the one in SRPA \\cite{Yannouleas1987}.\nAnalytical comparison of this correlation with the one in TBA \\cite{Tselyaev2007,Litvinova2010,Litvinova2013} is also given.\nThe framework has been used to study the isoscalar giant monopole, dipole, and quadrupole resonances of $^{16}$O using Skyrme functionals.\nThe results are compared with the second RPA \\cite{Gambacurta2010,Gambacurta2015} and (relativistic) time-blocking approximation \\cite{Lyutorovich2015,Litvinova2010,Litvinova2013}.\n\nThe importance of including self-consistently the full interaction in the PVC vertex has been shown, by considering the strength distributions and sum rules.\nAmong the different terms of the Skyrme interaction other than the central term, the spin-orbit term, which has been ignored in most previous PVC studies, plays a significant role in our current study, especially in the case of the ISGQR.\n\nThe diagonal approximation has also much influence on the strength distribution of the ISGQR in $^{16}$O.\nWithout diagonal approximation, the strength distribution of the ISGQR is more fragmented and wider, in better agreement with the experimental data.\nA new peak near $E = 15$ MeV appears in the PVC calculation without diagonal approximation (also present at lower energies in the experimental data). Such peak has been used as an example to show the difference induced by the diagonal approximation for the eigenenergies.\nFor the case of ISGMR and ISGDR, the strength distributions in $^{16}$O are less influenced by the diagonal approximation; and in \nall cases, the sum rules are not influenced by the diagonal approximation.\n\nAnother important drawback of the diagonal approximation is that one implicitly neglects the possibility of coupling between neutron and proton 1p-1h excitations included in the doorway states, that is instead recovered in the PVC calculation without diagonal approximation.\nThis difference is more prominent in situations where two phonons are dominated, respectively, by either a neutron or a proton 1p-1h excitation, as the interaction between 1p-1h excitations with different charge in the diagonal approximation is set to zero.\nWhen the phonon is already composed with mixed neutron and proton 1p-1h excitations, removing the diagonal approximation may not be significant.\n\nThe subtraction method, which has been developed to renormalize the effective interaction beyond RPA, has also been investigated within the framework of PVC calculations without diagonal approximation.\nIt solves, to some extent, the problem that the centroid of strength distributions is slightly too low compared with experimental data.\n\nAs can be seen from the formulas of the full spreading term in Eqs.~(\\ref{eq:wphj}), present PVC calculation without diagonal approximation is very time consuming, especially for heavy nuclei where more ph configurations are to be considered.\nIn the future we will make the calculation parallelized and apply it to study heavier systems.\n\nAlthough we have shown that removing the diagonal approximation is a step to be done, there is still room to improve the PVC models.\nWe plan to perform further investigation on the proper treatment of phonons in the doorway states.\nA recent investigation within the time blocking approximation \\cite{Tselyaev2018} might provide some interesting insight in this respect, as the authors propose a way to choose the most relevant phonons \nand achieve convergence with respect to the model space.\nThe works of \\cite{Litvinova2010,Litvinova2013,Litvinova2018a} are also of particular interest as a guidance for future development.\nOne can first take into account the ground-state correlation and include the RPA phonon coupling among the doorway states (see discussion in Appendix \\ref{app:tba}).\nFurther developments to compare with the equation-of-motion diagrams in \\cite{Litvinova2018a} and to go beyond 2p-2h level can also be made.\nAt the same time, the diagonal approximation should be tested in more nuclei and, even more importantly, in the case of other types of resonances such as spin-isospin resonances.\nThis may impact on the problem of the Gamow-Teller quenching or on the $\\beta$-decay processes\nof astrophysical interest.\n\n\\section*{ACKNOWLEDGMENTS}\n\nThis work was partly supported by Funding from the European Union's Horizon 2020 research and innovation programme under Grant agreement No. 654002.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}