{"text":"\\section{Introduction}\nThe interaction of protostellar outflows with the ambient molecular cloud occurs through radiative\nshocks that compress and heat the gas, which in turn cools down through line emission at\ndifferent wavelengths. In the dense medium where the\nstill very embedded protostars (the so called class 0 sources) are located, shocks are primarily non-dissociative, and hence the cooling is mainly through emission from \nabundant molecules. Molecular hydrogen is by far the most abundant species in these environments, \nand although H$_2$\\, emits only through quadrupole transitions with low radiative rates,\nit represents the main gas coolant in  flows from young protostars. H$_2$\\, shocked emission in outflows has been widely studied in the past mainly through its\nro-vibrational emission in the near-IR (e.g. Eisl{\\\"o}ffel et al. 2000; Giannini et al. 2004; Caratti o Garatti et al. 2006) that traces the dense\ngas at T$\\sim$2000-4000 K. Most of the thermal energy associated with the shocks is however radiated away through\nthe emission of H$_2$\\ rotational transitions of the ground state vibrational level at $\\lambda \\le 28\\mu$m\n(e.g. Kaufman \\& Neufeld 1996). \nMid-IR H$_2$\\, lines are easily excited at low densities and temperatures between 300 and 1500 K: therefore they\nare very good tracers of the molecular shocks associated with the acceleration of ambient gas by\nmatter ejection from the protostar. Given the low excitation temperature, \nthey can also probe regions where H$_2$\\, has not yet reached the ortho-to-para equilibrium, thus \ngiving information on the thermal history of the shocked gas (Neufeld et al. 1998; Wilgenbus et al. 2000).\nIn addition, given the different excitation temperature and critical densities of the v=0--0 and v$\\ge$ 1 H$_2$\\ lines,\nthe combination of mid-IR with near-IR observations is a very powerful tool to constrain \nthe global physical structure and the shock conditions giving rise to the observed emission.\n\nThe study of the 0--0 rotational emission in outflows started in some detail with the \n\\emph{Infrared Space Observatory}. Thanks to the observations performed with the SWS and ISOCAM instruments,\nthe shock conditions, the ortho-para ratio and the global H$_2$\\, cooling  have been derived in a handful of \nflows (e.g. Neufeld et al. 1998; Nisini et al. 2000; Molinari et al. 2000;  Lefloch et al. 2003). More recently, the \\emph{Infrared Spectrometer} (Houck et al. 2004) on board\n\\emph{Spitzer}, with its enhanced spatial resolution and sensitivity with respect to the ISO spectrometers, \nhas been used to obtain detailed images of the H$_2$\\, rotational emission, from S(0) to S(7), of several\noutflows, from which maps of  important physical parameters, such as temperature, column density and o\/p\nratio, have been constructed (Neufeld et al. 2006; Maret et al. 2009;  Dionatos et al. 2010). \nIn this framework, Neufeld et al. (2009, hereafter N09)  have recently presented IRS spectroscopic maps observations of five young \nprotostellar outflows at wavelengths between 5.2 and 37$\\mu$m, and discussed their averaged physical properties \nand overall energetics. In all the flows, the H$_2$\\, S(0)-S(7) emission has been detected and contributes to more than 95\\%  of the \ntotal line luminosity in the 5.2-37$\\mu$m\\, range, while atomic emission, in the form  of FeII and SI fine structure lines, accounts for only the remaining $\\sim$5\\%. \n\nIn the present paper, we will analyse the H$_2$\\, line maps obtained by N09 towards the L1157 outflow,\nwith the aim of deriving the main physical conditions pertaining to the molecular gas and their variations \nwithin the flow. This in turn will give information on the thermal history of the flow and on how \nenergy is progressively transferred from the primary ejection event to the slow moving ambient gas. \n\nFor this first detailed analysis, L1157 has been chosen among the sources observed by N09 given its uniqueness as a very active and well studied flow at different wavelengths.   \nMore than 20 different chemical species have been indeed \ndetected in the shocked spots of this object (Bachiller \\& Perez Gutierrez 1997; Benedettini et al. 2007), some of them for the first time in outflows (e.g. HNCO, Rodr{\\'{\\i}}guez-Fern{\\'a}ndez et al. 2010, and complex organic molecules, Arce et al. 2008, Codella et al. 2009)\n, testifying for a rich shock induced chemistry. Warm H$_2$\\ shocked emission in L1157\nis also evidenced through near-IR maps (e.g. Davis \\& Eisl\\\"offel, 1995) and Spitzer-IRAC\nimages (Looney et al. 2007). The L1157\noutflow has been also recently investigated with the \\textit{Herschel Space Observatory}, showing\nto be very strong also at far-IR wavelengths (Codella et al. 2010, Lefloch et al. 2010,\nNisini et al. 2010).\n\nThe L1157 outflow extends about 0.7 pc in length. Its distance is uncertain and has been estimated between 250 and 440 pc. Here we will adopt D=440 pc for an easier comparison with other works.\nThe outflow is driven by a highly embedded, low mass class 0 source (L1157-mm or IRAS20386+6751) having $L_{bol} \\sim $ 8.3 $L_\\odot$ (Froebrich 2005). \nIt is a very nice example of an outflow driven by a precessing and pulsed jet, possessing an S-shaped structure and different cavities, whose morphology has been reproduced assuming that the outflow\nis inclined by $\\sim$80$^\\circ$ to the line of sight and the axis of the underlying jet precesses\non a cone of 6$^\\circ$ opening angle (Gueth et al. 1996). The episodic mass ejection events are\nevidenced by the presence, along the flow, of individual clumps that are symmetrically displaced \nwith respect to the central source.\nIt is therefore a very interesting target for a  study of the physical conditions pertaining to these active regions through an H$_2$\\ excitation analysis.\n\nThe paper is organized as follow: the observations and the main results are summarized in \\S 2. In \\S 3 \nwe describe the analysis performed on the H$_2$\\ images to derive maps of temperature, column density and \northo-to-para ratio. A more detailed NLTE analysis on individual emission peaks is also presented here, where the \nSpitzer data are combined with near-IR data to further constrain the excitation conditions.  The implications of these results for the shock conditions along the L1157  flow are discussed in \\S 4, together with an analysis of the global\nenergy budget in the flow.  A brief summary follows in $\\S 5$.\n\n\\section{Observations and results \\label{analysis}}\n\nObservations of the L1157 outflow were obtained in November 2007 with the IRS instrument, during Cycle 4 of the Spitzer mission . \nThe full IRS spectral range (5.2-36.5$\\mu$m) was observed with the Long-High (LH), Short-High (SH) (R $\\sim$ 600) \nand Short-Low (SL) (R between 64 and 128) modules. \nThe L1157 outflow region was covered through 5  individual IRS maps of $\\sim$ 1\\arcmin x1\\arcmin\\,  of size each, arranged along the outflow axis. Each map was obtained by stepping the IRS slit by half of its width in the direction perpendicular\nto the slit length. For the SH and LH modules the slit was stepped also parallel to its axis by 4\/5 (SL) and 1\/5 (LH)  of its length. \nDetails on the data reduction that generated the individual line maps from the IRS scans are given in N09. The final maps have been resampled to a grid of 2\\arcsec\\, spacing allowing a pixel by pixel comparison of maps obtained with the different IRS modules.\nMaps of the brightest detected lines as well as the full spectrum in a representative position are shown in  Fig.\\,7 and Fig.\\,12 of N09. As regards to H$_2$, all the pure rotational lines of the first vibrational levels, from S(0) to S(7),\n are detected at various intensity along the flow. Here we report, in Tab. \\ref{fluxes},  the H$_2$\\, brightness measured in a 20\\arcsec\\, FWHM Gaussian aperture towards different positions.\n\n Fig\\,1  shows the L1157 maps of the S(1) and S(2) lines while Fig.\\,2 displays the S(5) line with superimposed contours of the CO 2--1 emission from Bachiller et al. 2001.\n In the same figure, a map of the 2.12 $\\mu$m 1--0 S(1) line is also presented. The morphology of the 0--0 S(5) and 1--0 S(1) is very similar, with  peaks of mid-IR emission \nlocated at the near-IR knots from A to D, as identified by Davis \\& Eisl\\\"offel (1995). \n When compared with the CO map, the mid-IR H$_2$\\, emission appears to follow the curved chain of clumps (labelled as B0-B1-B2 and R0-R1-R, for the blue-shifted and red-shifted lobes, respectively) that also correspond to peaks of SiO emission, as resolved in interferometric observation by Gueth et al. (1998) and Zhang et al. (2000). \n The L1157 outflow  morphology has been suggested to delineate a precessing flow (Gueth et al. 1996), where the H$_2$\\, and SiO peak emission knots follow  the location of the actual working surface of the precessing jet and are thus associated with the youngest ejection episodes. \nDiffuse H$_2$\\ emission is also detected in the S(1)-S(2) maps, that delineates the wall of a cavity that connects the central source with both\nthe R0 and B0 clumps. Such a cavity has been recognized in the CO 1-0 interferometric maps and it is likely created by the propagation of large bow-shocks.\nThe S(1)-S(2) maps of Fig.\\,1 show  extended emission of H$_2$\\,  also in the SE direction (i.e. where the B2 clump is located) and in the eastern edge of the northern lobe, that also follow  quite closely the CO morphology: these regions at lower excitation might trace additional cavities created by an older ejection episode of the precessing jet.\n\n\\section{ H$_2$ Analysis }\n\\subsection{LTE 2D analysis of the rotational lines: maps of averaged parameters }\n\\label{sec:maps}\nWe have used the  H$_2$\\, line maps to obtain the 2D distribution of basic H$_2$\\, physical parameters, through the analysis \nof the rotational diagrams in each individual pixel. As described in N09, who analysed the global \nH$_2$\\, excitation conditions in L1157, the distribution of upper level column densities of the S(0)-S(7) lines as a function of their \nexcitation energy, does not follow a straight line, indicating that a  temperature stratification in the \nobserved medium exists. The exact form of this temperature stratification depends on the type of shock\nthe H$_2$\\, lines are tracing, as they probe the post-shock regions where the gas cools from $\\sim  $ 1000 K \nto $ \\sim $ 200 K. \n\nThe simplest way to parametrize the post-shock temperature stratification\nis to assume a power-law distribution: this approach was applied by \nNeufeld \\& Yuan (2008), which also show that this type of distribution is expected  \nin gas ahead of unresolved bow-shocks. On this basis, and following also N09, we fit the observations \nassuming a slab of gas where the H$_2$\\, column density in each layer at a given $T$ varies as \n$ dN \\propto T^{-\\beta}dT$.\n\nThis law is integrated, to find the total column density, between a minimum ($T_{min}  $) and a maximum ($T_{max}$) temperatures . For our calculations \nwe have kept $T_{max}$ fixed at 4000 K, since gas at temperatures larger than this value is not expected to contribute to the emission of the observed pure rotational lines. $T_{min} $ was instead assumed to be equal, in each position, to the minimum temperature probed by the observed lines. This $T_{min} $ is taken as the excitation temperature giving\nrise to the observed ratio of the S(0) and S(1) column densities, assuming a Boltzman\ndistribution. \nThe $T_{min}  $ value ranges between $\\sim$150 and 400 K.\n\nWe found that the approach of a variable  $T_{min}$\nproduces always fits with a better $\\chi^2$ than assuming a fixed low value in all positions.\nWe also assume the gas is in LTE conditions. Critical densities of rotational lines from S(0) to S(7)\nrange between 4.9 cm$^{-3}$\\, (S(0)) and 4.4$\\times$10$^5$cm$^{-3}$(S(7)) at T=1000 K assuming only H$_2$\\, collisions\n(Le Bourlot et~al. 1999): critical densities decrease if collisions with H and He are not negligible. \nDeviations from LTE can be therefore expected only for the high-$J$ S(6) and S(7) transitions: the S\/N of these \ntransitions in the individual pixels is however not high enough to disentangle, in the rotational diagrams, NLTE \neffects from the effects caused by the variations of the other considered parameters. In particular, as also\ndiscussed in  N09, there is a certain degree of degeneracy in the density and the $\\beta$ parameter \nof the temperature power law in a NLTE treatment that we are not able to remove in the analysis of the\nindividual pixels. This issue will be further discussed in \\S \\ref{sec:NLTE}. \nAn additional parameter of our fit is the ortho-to-para ratio (OPR) value. It is indeed recognized that the 0--0 H$_2$\\, lines are\noften far from being in ortho-to-para equilibrium, an effect that in a rotational diagram is evidenced by\na characteristic zigzag behavior in which column densities of lines with even-$J$ lie systematically above those of odd-$J$ lines. In order not to introduce too many parameters, we assume a single OPR value as a free parameter for the\nfit. In reality, the OPR value is temperature dependent (e.g. N09 and  \\S \\ref{sec:NLTE}), and therefore\nthe high-$J$ lines  might present an OPR value closer to equilibrium then the low-$J$ transitions. \nOur fit gives therefore only a value averaged over the temperature range probed by the lines considered (i.e. $\\sim$200-1500 K). \n\nIn summary, we have varied only three parameters, namely the total H$_2$\\, column density N(H$_2$), the OPR, and the temperature power law index $\\beta$,  in order to obtain the best model fit through a $\\chi^2$ minimization procedure and assuming a 20$\\%$ flux uncertainty for all the lines. \nThe fit was performed only in those pixels where at least four lines with an S\/N larger\nthan 3 have been detected.  \nBefore performing the fit, the line column densities were corrected for extinction assuming $A_v$ = 2 (Caratti o Garatti et al. 2006) and \nadopting the Rieke \\& Lebofsky (1985) extinction law. At the considered wavelengths,\nvariations of $A_v$ of the order of 1-2mag do not affect any of the derived results.\n\nFigure \\ref{fig:ncol} shows the derived map of the H$_2$\\, column density, while in Fig. \\ref{fig:b_tmin} and \\ref{fig:op_tcold}  \nmaps of the OPR and  $\\beta$ are displayed together with temperature maps relative to the ''cold'' and ''warm'' components ($T_{cold}$ and $T_{warm}$),\ni.e. the temperature derived from linearly fitting the S(0)-S(1)-S(2) and S(5)-S(6)-S(7) lines, once corrected for the derived OPR value. \nIn Fig. 6 we also show the individual excitation diagrams for selected positions along the flow, obtained from intensities\nmeasured in a 20$\"$ FWHM Gaussian aperture centered towards emission peaks (Tab. 1). Values of the\nfitted parameters in these positions are reported in Tab. \\ref{param}. In addition to $T_{cold}$ and $T_{warm}$,\nwe give in this table also the values of the average temperature in each knot, derived through a\nlinear fit through all the H$_2$\\ lines ($T_{med}$).\n\nThe maps show significant variations in the inferred parameters along the outflow. \nThe H$_2$\\ column density ranges between 5$\\times$10$ ^{19} $ and \n3$\\times$10$ ^{20} $cm$^{-3}$. The region at the highest column density is located towards the B1 molecular bullet (see Fig. \\ref{fig:h2co})\\footnote{Fig. \\ref{fig:h2co} shows that the molecular clumps B1, R0 and\nR coincide in position with the NIR H$_2$\\ knots A, C and D. In the paper, both nomenclature\nwill be used specifying if we refer to the NIR or mm condensations}. \nThis is consistent with the higher column density of CO found in B1 with respect \nto other positions in the blue lobe (Bachiller \\& Perez Gutierrez 1997), and might suggest that this is a zone where the outflowing gas is compressed due \nto the impact with a region of higher density (Nisini et al. 2007). Towards the NW, red-shifted \noutflow, the column density has a more uniform distribution, with a plateau at $ \\sim 10 ^{20} $ cm$^{-2}$\\ that follows the H$_2$\\ intensity distribution. \nThe N(H$_2$) decreases at the apex of the red-shifted outflow, with a value slightly below $ 10 ^{20} $ cm$^{-2}$\\ at the position\nof the D near-IR knot.\n\n$T_{cold}$ ranges between $ \\sim $ 250 and 550 K. The highest values are found at the tip of the northern outflow lobe, while local maxima corresponds to the positions of line intensity peaks.\n$T_{warm}$ ranges between  $ \\sim $ 1000 and 1500 K. In this case the highest values are in the southern lobe, at the position of the A NIR knot. \nAs a general trend, the $T_{warm}$ value decreases going from the southern to the northern peaks of emission, with\nthe minimum value at the position of the D NIR knot. \n\n$\\beta$ values range between $ \\sim $4-4.5 in the blue-shifted lobe while it is larger in the red-shifted\nlobe, with maximum values of $ \\sim $ 5.5 at the tip of the flow. \nDue to the degeneracy between $\\beta$ and density discussed in the previous section, these\nvalues can be considered as upper limits because of our assumption of LTE conditions. \nNeufeld \\& Yuan (2008) have discussed the $\\beta$ index expectations in bow-shock excitation.\nA $\\beta$ index of $ \\sim $ 3.8 is expected in paraboloid bow shocks having a velocity at the bow apex\nhigh enough to dissociate H$_2$,  in which case the temperature distribution\nextends to the maximum allowed temperature. Slower shocks that are not able to attain the maximum\ntemperature, produce steeper temperature distributions, i.e. with values of $ \\beta $ greater than 3.8.\nThis is consistent with our findings: low values of $ \\beta $ (of the order of 4) are found in the blue-shifted\nlobe: here, evidence of H$_2$\\ dissociation is given by the detection\nof atomic lines (i.e. [SiII], [FeII] and [SI]) in the IRS spectrum. In the red-shifted lobe, where values of $\\beta$ larger than 4 are derived in the LTE\nassumption, no atomic emission is detected and the $T_{warm}$ values\nare lower than those measured in the blue-shifted lobe, indicating a maximum temperature lower than in the blue flow .\n\nThe OPR varies significantly along the outflow, spanning from $ \\sim $ 0.6 to 2.8. Hence it is always below the equilibrium value of 3.  Although a one-to-one correlation between temperature and OPR cannot be discerned, some trends can be\ninferred from inspection of Fig. \\ref{fig:op_tcold}. In general the OPR minima are observed in plateau regions between two consecutive\nintensity peaks, where also the cold temperature has its lowest values. At variance with this trend, the emission\nfilaments delineating the outflow cavity within $\\pm$ 20$ \\arcsec  $ from the mm source, where the cold \ntemperature reaches a minimum value of $ \\sim $ 250 K,   show rather high\nvalues of the OPR, $ \\sim $2.4-2.8. This might suggests that this region has experienced an older \nshock event that has raised the OPR, though not to the equilibrium value, and where the gas\nhad time to cool at a temperature close to the pre-shock gas temperature. \nOn the other hand, at the apex of both  the blue- and red-shifted lobes, where the cold temperatures \nare relatively high (i.e. 500-550 K), the OPR is rather low, $ 1.5-2.0 $. Evidence of regions of low OPR and high \ntemperatures at the outflow tips has been already given in other flows (Neufeld et al. 2006; Maret et al. 2009). It has been suggested that these represent zones subject to recent shocks where the OPR has not had time yet to reach the\nequilibrium value. \n\n\n\\subsection{ NLTE analysis: constraints on H and H$_2$ particle density}\n\\label{sec:NLTE}\n\nAdditional constraints on the physical conditions responsible for the H$_2$\\, excitation, are provided by combining\nthe emission of the mid-IR H$_2$\\ pure-rotational lines from the ground vibrational level with the emission from near-IR ro-vibrational lines.\nIt can be seen from Fig. \\ref{fig:h2co} that the 2.12$\\mu$m\\, emission follows quite closely the\nemission of the 0--0 lines at higher $J$. In addition to the 2.12$\\mu$m\\, data presented in Fig. \\ref{fig:h2co}, we have also considered\nthe NIR long-slit spectra obtained on the A and C NIR knots by Caratti o Garatti (2006). These knots, at the spatial\nresolution of the NIR observations (i.e. $\\sim$0.8\\arcsec), are separated in several different sub-structures that have been individually investigated with the long-slit spectroscopic observations. For our analysis we have considered the\ndata obtained on the brightest of the sub-structures,  that coincide, in position, \nwith peaks of the 1--0 S(1)  line.\n\nIn order to inter-calibrate in flux the Spitzer data with these NIR long-slit data, obtained  with a slit-width of 0.5 arcsec, \nwe have proceeded as follows: we first convolved the \n2.12$\\mu$m\\, image at the resolution of the Spitzer images and than performed photometry on the\nA and C peaks positions with a 20\\arcsec\\, diameter Gaussian aperture, i.e. with the same\naperture adopted for the brightness given in Tab. \\ref{fluxes}. We have then scaled the fluxes of the individual lines given in \nCaratti o Garatti (2006) in order to match the 2.12$\\mu$m\\ flux gathered in the slit with that measured by the\nimage photometry. In doing this, we assumed that the average excitation conditions within the 20$\\arcsec$ aperture are\nnot very different from those of the A-C peaks. This assumption is \nobservationally supported by the fact that the ratios of different H$_2$\\ NIR lines \ndo not change significantly (i.e. less than 20\\%) in the A-C knot substructures separately\ninvestigated in Caratti o Garatti (2006).\nWe have considered only those lines detected with S\/N larger \nthan 5; in practice this means considering lines from the first four and three vibrational levels for knots A and C, respectively. \nThe excitation diagrams obtained by combining the Spitzer and NIR data for these two knots are displayed in Fig.\\ref{fig:ACfit}.\n\nIn order to model together the 0--0 lines and the near-IR ro-vibrational lines, we have implemented\n two  modifications to the approach adopted previously. First of all, the NIR lines probe gas\nat temperatures higher than the pure rotational lines, of few thousands of K, at which values it is expected\nthat the OPR has already reached equilibrium. Thus, \nthe ortho-para conversion time as a function of temperature needs to be included in the\nfitting procedure, since lines excited at different temperatures have different OPRs.\nWe  have here adopted the approach of N09 and used an analytical expression for the OPR as a function of the temperature, considering a gas that had an initial value of the ortho-to-para ratio OPR$_0  $ and has been heated to a temperature T for a time $  \\tau$. Assuming that the para-to-ortho conversion occurs through reactive collisions with atomic hydrogen, we have:\n\n\\begin{equation}\n\\frac{OPR(\\tau)}{1+OPR(\\tau)}  = \\frac{OPR_0}{1+OPR_0}\\,e^{-n(H)k\\tau} + {\\frac{OPR_{LTE}}{1+OPR_{LTE}}}\\,\\left(  1 - e^{-n(H)k\\tau}\\right) \n\\end{equation}\n\nIn this expression, n(H) is the number density of atomic hydrogen and OPR$_{LTE}$ is the ortho-to-para ratio equlibrium value.\nThe parameter $ k $ is given by the sum of the rates coefficients for para-to-ortho conversion ($ k_{po} $), estimated\nas 8$ \\times $10$ ^{-11} $exp(-3900\/T) cm$ ^{3} $\\,s$ ^{-1} $, and for ortho-to-para conversion, $ k_{op} \\sim k_{po}\/3 $\n(Schofield et al. 1967). Thus the dependence of the OPR on the temperature is implicitly given by the dependence on T of the\n$ k $ coefficient. The inclusion of a function of OPR on $T$, introduces one additional parameter to our fit: while we have previously \nconsidered only the average OPR of the 0--0 lines, we will fit here the initial OPR$_0  $ value and the coefficient $ K = n(H)\\tau $.\n\nThe second important change that we have introduced with respect to the previous fitting procedure, \nis to include a NLTE treatment of the H$_2$\\, level column densities. In fact, the critical densities\nof the NIR lines are much higher than those of the pure rotational lines (see Le Bourlot et~al. 1999). For example, the \n$n _{crit} $ of the 1-0 S(1) 2.12$\\mu$m\\, line is 10$^7$cm$^{-3}$\\, assuming only collisions with H$_2$\\, and T=2000 K. \nTherefore the previously adopted LTE approximation might not be valid when combining lines from different \nvibrational levels.\nThis is illustrated in Fig.\\ref{fig:plot_dens}, where we plot the results obtained by varying the H$_2$\\ density between \n10$ ^{3} $ and 10$ ^{7} $cm$^{-3}$\\, while keeping the other model parameters fixed . The observed column densities in the A\nposition are displayed for comparison. \nFor the NLTE statistical equilibrium computation we have adopted the H$_2$\\, collisional rate coefficients given by Le Bourlot et~al. (1999) \\footnote{The  rate coefficients for collisions with ortho- and para-H$_2$, HI and He, computed in Le Bourlot et~al. (1999) \nare available at the web-site: http:\/\/ccp7.dur.ac.uk\/cooling$\\_$by$\\_$h2\/} .\nThis figure demonstrates the sensitivity of the relative ratios  between 0--0 and 1--0 transitions to density variations.   For example,  in this particular case, the ratio N(H$_2$)$_{0-0 S(7)}$\/N(H$_2$)$_{1-0 S(1)}$ is 64.6 at n(H$_2$)=10$ ^{4} $ cm$^{-3}$\\, and 1.9 at n(H$_2$) $ \\gtrsim $ 10$ ^{7} $ cm$^{-3}$. The figure also shows that the observational points display only a small misalignment in column densities between the 0--0 lines and the 1--0 lines, already indicating that the ro-vibrational lines are close to\nLTE conditions at high density.\n\nIn Fig.\\ref{fig:ACfit}, we show the final best-fit models for the combined mid- and near-IR column densities \nin the A and C positions. \nAs anticipated, the derived n(H$_2$) densities are large, of the order of 10$ ^{7} $ and 6$\\times$10$ ^{6}$ cm$^{-3}$,  for the A and C positions, respectively. The two positions indeed show very similar excitation conditions: only the column density is a factor of 3 smaller in knot C. \nHence, we conclude that the lack of detection of rotational lines with v$ > $ 3 in knot C, in contrast to knot A (Caratti o Garatti et al. 2006), is due\npurely to a smaller number of emitting molecules along the line of sight and not to different excitation conditions.\n\nThe derived H$_2$\\, densities are much higher than  previous estimates based on other tracers. Nisini et al. (2007)\nderive a density of 4$\\times$10$ ^{5} $ cm$^{-3}$\\, at the position of knot A from multi-line SiO observations, thus more than an order of magnitude smaller than those inferred from our analysis. The high-J CO lines observed along the blue-shifted lobe of L1157 by \nHirano et al. (2001), indicate a density even smaller, of the order of 4$\\times$10$ ^{4} $ cm$^{-3}$. \nSiO is synthesized and excited in a post-shock cooling zone where the maximum compression is reached, therefore\nit should trace post-shock regions at densities higher than H$_2$ (e.g. Gusdorf et al. 2008). \nOne possibility at the origin of the discrepancy is our assumption of collisions with only H$_2$\\ molecules, and thus of a \nnegligible abundance of H. This can be considered roughly true in the case of non-dissociative C-shocks, where H atoms are \nproduced primarely in the chemical reactions that form H$_2$O from O and H$_2$, with an abundance n(H)\/n(H$_2$+H) $ \\sim  $ 10$^{-3}$\n(e.g. Kaufmann \\& Neufeld 1996). However, if the shock is partially dissociative, the abundance of H can increase considerably and \ncollisions with atomic hydrogen cannot be neglected, in view of its  large efficiency in the H$_2$\\, excitation.\nThis situation cannot be excluded at least for the knot A, where atomic emission from [FeII] and [SI] has been \ndetected in our Spitzer observations.\nSince we cannot introduce the n(H) as an additional independent parameter of our fit, we have fixed n(H$_2$) at the value\nderived from SiO observations (4$\\times$10$ ^{5} $ cm$^{-3}$, Nisini et al. 2007) and varied the H\/H$_2$\\, abundance ratio. \nThe best fit is in this case obtained with a ratio H\/H$_2$=0.3: this indicates that our observational data are\nconsistent with previous H$_2$\\ density determinations only if a large fraction of the gas is in atomic form.\n\nTurning back to the inferred OPR variations with temperature, our fit implies that the OPR in the cold gas component at T=300 K\nis significantly below the equilibrium value, while the value of  \nOPR=3 is reached in the hot gas at T=2000 K traced by the NIR lines. The parameter  $K=n(H)\\tau$\nis constrained to be $\\sim$10$^6$ and 10$^7$ yr\\,cm$^{-3}$\\, for knots A and C, respectively. \nWe can also estimate the time needed for the gas to reach this distribution of OPR, from the limits on the\natomic hydrogen abundance previously discussed.\nOur data implies a high value of the n(H) density: a minimum value of n(H) $\\sim$0.6-1$\\times$10$^4$cm$^{-3}$, (for knots C and A, respectively) \n is given if we assume H\/H$_2$$\\sim$ 10$^{-3}$ (and thus the n(H$_2$) $\\sim$ 10$ ^{7} $cm$^{-3}$, given\n by our fit), while a maximum value of $\\sim$10$^5$cm$^{-3}$,\nis derived from the fit where n(H$_2$) is kept equal to 4$\\times$10$ ^{5} $ cm$^{-3}$.\nThe high abundance of atomic H ensures that conversion of para- to\northo-H$_2$\\, proceeds very rapidly: the fitted values of the K parameter indeed imply that the observed range of\nOPR as function of temperature have been attained in a timescale between 100 and 1000 yrs for both the knots.\n\nFinally, given the column density and particle density discussed above, we can estimate the H$_2$\\,\ncooling length ($L \\sim$ N(H$_2$)\/n(H$_2$)). If we consider the case of n(H$_2$) $\\sim$ 10$ ^{7} $cm$^{-3}$,\nand negligible n(H), we have $L \\sim$ 10$ ^{13} $ cm while a length of $\\sim$ 10$ ^{15} $ cm is inferred\nin the case of n(H$_2$)$\\sim$ 4$\\times$10$^5$cm$^{-3}$. \nAll the parameters derived from the above analysis are summarised in Tab. \\ref{shock} and they will be discussed\nin the next section in the framework of different shock models.\n\n\\section{Discussion}\n\n\\subsection{Shock conditions giving rise to the H$_2$\\, emission}\n\nThe copious H$_2$\\, emission at low excitation observed along the L1157 outflow \nindicates that the interaction of the flow with the ambient medium occurs\nprevalently through non-dissociative shocks. \nBoth the Spitzer IRS maps of N09, and the NIR narrow band images of Caratti o Garatti et al. 2006, show that significant gas dissociation in L1157 occurs only at the A peak, where both mid-IR \nlines from [FeII], [SII] and [SI] and weak [FeII] at 1.64$\\mu$m\\ have been detected. \nWeaker [SI]\\,25$\\mu$m\\ and [FeII]\\,26$\\mu$m\\ emission have been also detected on the C spot, but\noverall the atomic transitions give a negligible contribution to the total gas cooling,\nas pointed out in N09.\nThese considerations suggest that most of the shocks along the outflow occur at speeds\n below $\\sim$ 40km\\,s$^{-1}$,  as this is the velocity limit above which H$_2$\\ is expected to be dissociated.\nThe knot A is the only one showing a clear bow-shock structure. Here the velocity at\nthe bow apex is probably high, causing H$_2$\\ dissociation and atomic line excitation,\nconsistent with the fitted temperature power law $\\beta$ of $\\sim$ 4, as discussed in \\S 3.1,\nwhile the bulk of the H$_2$\\, emission comes from shocks at lower velocities originating in\nthe bow wings.\n\nConstraints on the shock velocity that gives rise to the molecular emission in L1157\n have been already given in previous works. The sub-mm SiO emission and\nabundances, measured in different outflow spots, suggest shock velocities of the \norder of 20-30km\\,s$^{-1}$\\ (Nisini et al. 2007). The comparison of SiO and H$_2$\\ \nemission against detailed shock models performed by Gusdorf et al. (2008) confirm a similar\nrange of velocities in the NIR-A knot, although the authors could not find a unique \nshock model that well represents both the emissions. \n\nCabrit et al. (1999) found that the column density of the mid-IR H$_2$\\, emission lines, \nfrom S(2) to S(8), observed by ISO-ISOCAM was consistent either with C-shocks having\nvelocities of $\\sim$\\,25km\\,s$^{-1}$\\, or with J-shocks at lower velocity, of the order\nof $\\sim$\\,10km\\,s$^{-1}$. Gusdorf et al. (2008), however, conclude\nthat stationary shock models, either of C- or J-type, are not able to reproduce the observed\nrotational diagram on the NIR-A position, constructed combining ISOCAM data and \nNIR vibrational lines emission. A better fit was obtained \nby these authors considering non-stationary shock models, which have developed a magnetic precursor but which retain a J-type discontinuity (the so-called CJ shocks, Flower et al. 2003). \nSimilar conclusions, but on a different outflow, have been reached by Giannini et al. 2006\nwho studied the H$_2$\\ mid- and near-IR emission in HH54: in general, stady-state C- and J-type \nshocks fail to reproduce simultaneously the column densities of both the ro-vibrational \nand the v=0, pure-rotational H$_2$\\ levels.  \n\nA different way to look at the issue of the prevailing shock conditions in the observed\nregions, is to compare the set of physical parameters that we have inferred from our analysis \nto those expected from different shocks. With this aim, we summarize in   \nTab. \\ref{shock} the physical properties derived on the A and C H$_2$\\, knots. \nIn addition to the parameters derived from the NLTE analysis \n reported in Section \\ref{analysis}, namely H$_2$ post-shock density, H\/H$_2$\\ fraction, \n initial OPR, cooling length and time, the table reports also the average values of \nOPR and rotational temperature, as they are measured from a simple linear fit\n of the rotational diagrams presented in Fig. \\ref{fig:fit_nir}.\n\n\nAs mentioned in \\S 3.2, the high fraction of atomic hydrogen inferred by our analysis rule out excitation in a pure\n C-shock. In fact, dissociation in C-shocks is always too low to have a \nH\/H$_2$ ratio higher than 5$\\times$10$^{-3}$, irrespective from the shock velocity and magnetic field strength \n(Kaufman \\& Neufeld 1996; Wilgenbus et al. 2000).\nC-shocks are not consistent with the derived parameters even if we consider the model fit with\nthe high H$_2$\\ post-shock density of the order of 10$^{7}$ cm$^{-3}$ and negligible atomic hydrogen: in this case we derive an emission length of 10$^{13}$ cm,\nwhich is much lower than the cooling length expected in C-shocks, which, although \ndecreasing with the pre-shock density, is never less than 10$^{15}$ cm (Neufeld et al. 2006) .\n\nStationary J-shock models better reproduce some of our derived parameters.\nFor example, in J-type shocks the fraction of hydrogen in the post-shocked gas\ncan reach the values of 0.1-0.3 we have inferred, provided that the shock velocity is larger\nthan $\\sim$ 20 km\\,s$^{-1}$. In general, a reasonable agreement with the inferred post-shock density and \nH\/H$_2$\\ ratio is achieved with models having $v_s$=20-25 km\\,s$^{-1}$ and pre-shock densities of 10$^3$cm$^{-3}$\n(Wilgenbus et al. 2000). Such models predict a shock flow time of the order of 100 yr or less, which\nis also in agreement with the value estimated in our analysis at least in knot A.\n In such models, however, the cooling length is an order of magnitude smaller than the \ninferred value of $ \\sim $10$ ^{15} $ cm. In addition, the gas temperature remains high for most of the post-shocked region: the average rotational temperature of \nthe v=0 vibrational level is predicted to be, according to the Wilgenbus et al. (2000)\ngrid of models, always about 1600 K or larger, as compared with the value of about 800-900 K inferred from observations. \nThe consequence of the above inconsistencies is that J-type shocks tend to underestimate \nthe column densities of the lowest H$_2$\\, rotational levels in L1157, an effect already pointed \nout by Gusdorf et al. (2008). \n\nAs mentioned before, Gusdorf et al. (2008) conclude that the H$_2$\\ pure rotational emission in L1157 \nis better fitted with a non-stationary C+J shock model with either $v_s$ between 20 and 25 km\\,s$^{-1}$\\ and pre-shock densities $n_H = 10^4$ cm$^{-3}$, or with $v_s \\sim 15$ km\\,s$^{-1}$\\, and\nhigher pre-shock densities of $n_H = 10^5$ cm$^{-3}$. Such models, however, still underestimate\nthe column densities of the near-IR transitions: the post-shocked H$_2$\\ gas density  \nremains lower than the NIR transitions critical density and the atomic hydrogen\nproduced from H$_2$\\ dissociation is not high enough to populate the vibrational\nlevels to equilibrium conditions.\n\nThe difficulty of finding a suitable single model that reproduce the derived physical \nconditions is likely related to possible geometrical effects and to the fact that multiple shocks with different velocities might be present along the line of sight. It would be indeed  interesting to explore whether bow-shock models might be able to predict the averaged physical characteristics \nalong the line of sight that we infer from our analysis.\n\n\\subsection{Flow energetics}\n\nH$_2$\\ emission represents one of the main contributor to the energy radiated\naway in shocks along outflows from very young stars. Kaufman \\& Neufeld (1996) predicted that\nbetween 40 and 70\\% of the total shock luminosity is emitted in H$_2$\\, lines\nfor shocks with pre-shock density lower than 10$^5$ cm$^{-3}$\\ and shock velocities larger\nthan 20 km\\,s$^{-1}$, the other main contributions being in CO and H$_2$O rotational emission. \nThis has been also observationally tested by Giannini et al. (2001) who measured the \nrelative contribution of the different species to the outflow cooling in a sample of class 0 \nobjects observed with ISO-LWS. \n\nWe will discuss here the role of the H$_2$\\, cooling in the global radiated energy of the L1157 outflow.\nFrom the best fit model obtained for the knots A and C, we have derived the total, extinction corrected,\nH$_2$\\ luminosity by integrating over all the ro-vibrational transitions considered by our\nmodel. $L_{\\rm H_2}$ is found\nto be 8.4$\\times$10$^{-2}$ and 3.7$\\times$10$^{-2}$ L$_\\odot$ for the A and C knots, respectively. Out of this total\nluminosity, the contribution of only the rotational lines is 5.6$\\times$10$^{-2}$(A) and 2.7$\\times$10$^{-2}$(C) L$_\\odot$,\nwhich means that in both cases they represent about 70\\% of the total H$_2$\\, luminosity.\n\nN09 have found that the total luminosity of the H$_2$\\ rotational lines from S(0) to\nS(7), integrated over the entire L1157 outflow, \namount to 0.15 L$_\\odot$. If we take into account an additional 30\\% of contribution from the v$>$0 \nvibrational levels, we estimate a total H$_2$\\ luminosity of 0.21 L$_\\odot$. This is a 30\\% larger\nthan the total H$_2$\\ luminosity estimated by Caratti o Garatti (2006) in this outflow, \nassuming a single component gas at temperature between 2000 and 3000 K that fit the NIR H$_2$\\ lines. \n\nIf we separately compute the H$_2$\\ luminosity\nin the two outflow lobes, we derive $L_{\\rm H_2}$ = 8.5$\\times$10$^{-2}$ L$_\\odot$ in the blue lobe and 1.3$\\times$10$^{-1}$ L$_\\odot$ in the red\nlobe. Comparing these numbers with those derived in the individual A and C knots, \nwe note that the A knot alone contributes to most of the H$_2$\\ luminosity in the blue lobe. By contrast,\nthe H$_2$\\ luminosity of the red lobe is distributed among several peaks of similar values. \nThis might suggest that most of the energy carried out by the blue-shifted jet is\nreleased when the leading bow-shock encounters a density enhancement at the position of the A knot. \nOn the other hand, the red-shifted gas flows more freely without large density discontinuities, and \nthe corresponding shocks are internal bow-shocks, all with similar luminosities.\n\nThe integrated luminosity radiated by CO, H$_2$O and OI in L1157 has been estimated, through ISO and\nrecent Herschel observations, as $\\sim$0.2 L$_\\odot$ (Giannini et al. 2001, Nisini et al. 2010), \nwhich means that H$_2$\\ alone contributes about 50\\% of the\ntotal luminosity radiated by the outflow. \nIncluding all contributions, the total shock cooling along the L1157 outflow amounts to about 0.4 L$_\\odot$, i.e. $L_{cool}\/L_{bol}$ $\\sim$ 5$\\times$10$^{-2}$, assuming $L_{bol}$=8.4 L$_\\odot$ \nfor L1157-mm (Froebrich 2005). This ratio is consistent with the range of values derived from other class 0 sources \nfrom ISO observations (Nisini et al. 2002).\n\nThe total kinetic energy of the L1157 molecular outflow \nestimated by Bachiller et al. (2001) amounts to 0.2 L$_\\odot$ without any correction for the\noutflow inclination angle, or to 1.2 L$_\\odot$ if an inclination angle of 80 degrees \nis assumed. Considering that the derivation of the L$_{kin}$ value has normally\nan uncertainty of a factor of five (Downes \\& Cabrit 2007), we conclude that the mechanical energy flux \ninto the shock, estimated as $L_{cool}$,  is comparable to the kinetic energy of the swept-out \noutflow and thus that the shocks giving rise to the H$_2$ emission have \nenough power to accelerate the molecular outflow.\n\nThe total shock cooling derived above can be also used to infer the momentum flux \nthrough the shock, i.e. $\\dot{P}$ = 2$L_{cool}$\/V$_s$, where V$_s$ is the shock velocity that\nwe can assume, on the basis of the discussion in the previous section, to be of the order of 20 km\\,s$^{-1}$.\nComputing the momentum flux separately for the blue and red outflow lobe, we derive \n$\\dot{P}_{red} \\sim$ 1.7$\\times$10$^{-4}$ and $\\dot{P}_{blue} \\sim$ 1.1$\\times$10$^{-4} $M$_\\odot$ yr$^{-1}$ km\\,s$^{-1}$. \nIn this calculation, we have assumed that the contribution from cooling species\n different from H$_2$, as estimated by ISO and Herschel, is distributed among the two lobes \n in proportion to the H$_2$\\ luminosity. If we assume that the molecular \n outflow is accelerated at the shock front through momentum conservation, then the\n above derived momentum flux should results comparable to the thrust of the outflow,\n derived from the mass, velocity and age measured through CO observations.\nThe momentum flux measured in this way by Bachiller et al. (2001) is 1.1$\\times$10$^{-4}$and 2$\\times$10$^{-4}$ M$_\\odot$ yr$^{-1}$ km\\,s$^{-1}$\nin the blue and red lobes, respectively, i.e. comparable to our derived values. It is interesting to note that \nthe $\\dot{P}$ determination from the shock luminosity confirms the asymmetry between\nthe momentum fluxes derived in two lobes. As shown by Bachiller et al. (2001), the L1157 red lobe \nhas a 30\\% smaller mass with respect to the blue lobe, but a higher momentum flux due to the larger flow velocity. The northern red lobe is in fact more extended than the southern lobe:\nhowever, given the higher velocity of the red-shifted gas, the mean kinematical ages of the two lobes is very\nsimilar. \n\n\n\\section{Conclusions \\label{conclusions}}\n\nWe have analysed the H$_2$\\ pure rotational line emission, from S(0) to S(7),\nalong the outflow driven by the L1157-mm protostar, mapped with the Spitzer - IRS \ninstrument. The data have been analysed assuming a gas temperature stratification where the H$_2$\\ column \ndensity varies as $T^{-\\beta}$ and 2D maps of the H$_2$\\ column density,\northo-to-para ratio (OPR) and temperature spectral index $\\beta$\nhave been constructed. \nFurther constraints on the physical conditions of the shocked gas have been derived \nin two bright emission knots by combining the Spitzer observations with near-IR \ndata of H$_2$\\ ro-vibrational emission. Finally, the global H$_2$\\ radiated energy of the\noutflow has been discussed in comparison with the energy budget of the associated\nCO outflow.\n\nThe main conclusions derived by our analysis are the following:\n\\begin{itemize}\n\\item H$_2$\\ transitions with $J_{lower} \\le$ 2 follows the morphology of the CO molecular\noutflow, with peaks correlated with individual CO clumps and more diffuse\nemission that delineates the CO cavities created by the precessing jet. \nLines with higher $J$ are localized on the shocked peaks, presenting a morphology\nsimilar to that of the H$_2$\\, 2.12$\\mu$m\\, ro-vibrational emission.\n\\item  Significant variations of the derived parameters are observed along the flow. \nThe H$_2$\\ column density ranges between 5$\\times$10$^{19} $ and 3$\\times$10$^{20} $cm$^{-2}$: \nthe highest values are found in the blue-shifted lobe, suggesting that here the outflowing\ngas is compressed due to the impact with a high density region.\nGas components in a wide range  of temperature values, from $ \\sim $ 250 to $ \\sim $ 1500 K\ncontribute to the H$_2$\\ emission along individual lines of sight. The largest range\nof temperature variations is derived towards the intensity peaks closer to the \ndriving source, while a more uniform temperature distribution, with $ T $\nbetween $ 400 $ and  $ 1000$ K, is found at the tip of the northern outflow lobe.\n\\item The OPR is in general lower than the equilibrium value at high temperatures and spans a range from $\\sim$0.6 to 2.8, with the lowest values\nfound in low temperature plateau regions between consecutive intensity peaks. \nAs in previous studies, we also found the presence of regions at low OPR (1.5-1.8) \nbut with relatively high temperatures. These might represent zones subject to recent shocks \nwhere the OPR has not had time yet to reach the equilibrium value.\n\n\\item Additional shock parameters have been derived in the two bright near-IR \nknots A and C,  located \nin the blue- and red-shifted outflow lobes, where the mid- and near-IR H$_2$\\ \ndata have been combined. \nThe ratio between mid- and near-IR lines is very sensitive to the molecular  plus atomic hydrogen particle density. A high \nabundance of atomic hydrogen (H\/H$_2$ $\\sim$ 0.1-0.3) is implied by the \nthe observed H$_2$\\ column densities if we assume n(H$_2$) values as derived by independent \nmm observations. With this assumption, the cooling lengths  of the shock result \nof the order of 7$\\times$10$ ^{14} $  and 10$ ^{15} $ cm for the A and C knot, respectively.\nThe distribution of OPR values as a function of temperature  and the \nderived abundance of atomic hydrogen, implies that the shock passing time is of the\norder of 100 yr for knot A and 1000 yr  for  knot C, given the assumption that the para-to-ortho\nconversion occurs through reactive collisions with atomic hydrogen.\nWe find that planar shock models, either of C- or J-type, are\nnot able to consistently reproduce all the physical parameters derived from our analysis \nof the H$_2$\\ emission. \n\\item Globally, H$_2$\\ emission contributes to about 50\\% of the total shock radiated energy in the L1157 outflow. We find that the momentum flux through the shocks derived from the radiated luminosity is\ncomparable to the thrust of the associated molecular outflow, supporting a scenario\nwhere the working surface of the shocks drives the molecular outflow. \n\\end{itemize}\n\n\\acknowledgments\n\nThis work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Financial support from contract ASI I\/016\/07\/0 is acknowledged. \n\n\\bibliographystyle{plainnat}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nFrames are overcomplete (or redundant) sets of vectors that serve to faithfully represent signals. They were introduced in $1952$ by Duffin and Schaeffer \\cite{duscha52}, and reemerged with the advent of wavelets \\cite{Christensen:2003ab,  Dau92,Ehler:2007aa, Ehler:2008ab, hw89}. \nThough the overcompleteness of frames precludes signals from having unique representation in the frame expansions, it is, in fact, the driving force behind the use of frames in signal processing \\cite{Casazza:2003aa, koche1, koche2}.\n\nIn the finite dimensional setting, frames are exactly spanning sets. However, many applications require ``custom-built'' frames that possess additional properties which are dictated by these applications. As a result, the construction of frames with prescribed structures  has been actively pursued. For instance, a special class called {\\it finite unit norm tight frames } (FUNTFs) that provide a Parseval-type representation very similar to orthonormal bases, has been customized to model data transmissions \\cite{Casazza:2003aa, Goyal:2001aa}. Since then the characterization and construction of FUNTFs and some of their generalizations have received a lot of attention \\cite{Casazza:2003aa, koche1, koche2}. Beyond their use in applications, FUNTFs are also related to some deep open problems in pure mathematics such as the Kadison-Singer conjecture \\cite{cftw}. FUNTFs appear also in statistics where, for instance,  Tyler used them to construct $M$-estimators of multivariate scatter \\cite{Tyler:1987}. We elaborate more on the connection between the $M$-estimators and FUNTFs in Remark~\\ref{remark:M estimator FUNTF}. These $M$-estimators were subsequently used to construct maximum likelihood estimators for the the wrapped Cauchy distribution on the circle in \\cite{KentTaylor1994} and for the angular central Gaussian distribution on the sphere in \\cite{Tyler:1988}. \n\nFUNTFs are exactly the minimizers of a functional called the frame potential \\cite{Benedetto:2003aa}. This was extended to characterize all finite tight frames in \\cite{Waldron:2003aa}. Furthermore, in \\cite{fjko, jbok}, finite tight frames with a convolutional structure, which can be used to model filter banks, have been characterized as minimizers of an appropriate potential. All these potentials are connected to other functionals whose extremals have long been investigated in various settings. We refer to \\cite{cokum06, Delsarte:1977aa, Seidel:2001aa, Venkov:2001aa, Welch:1974aa} for details and related results. \n\nIn the present paper, we study objects beyond both FUNTFs and the frame potential. In fact, we consider a family of functionals, the {\\it $p$-frame potentials}, which are defined on  sets $\\{x_{i}\\}_{i=1}^{N}$ of unit vectors in $\\R^d$; see Section~\\ref{section:pfp}. These potentials have been studied in the context of spherical $t$-designs for even integers $p$, cf.~Seidel in \\cite{Seidel:2001aa}, and their minimizers are not just FUNTFs but FUNTFs that inherit additional properties and structure. Common FUNTFs are recovered only for $p=2$. In the process, we extend Seidel's results on spherical $t$-designs in \\cite{Seidel:2001aa} to the entire range of positive real $p$. \n\nIn Section~\\ref{section:estimates}, we give lower estimates on the $p$-frame potentials, and prove that in certain cases their minimizers are FUNTFs, which possess additional properties and structure. In particular, if $0<p \\leq 2$, we completely characterize the minimizers of the $p$-frame potentials when $N=kd$ for some positive integer $k$. Moreover, when $N=d+1$ and $0<p\\leq 2$, we characterize the minimizers of the $p$-frame potentials, under a technical condition, which, we have only been able to establish when $d=2$. We conjecture that this technical condition holds when $d>2$. Finally in Section \\ref{section:intro prob}, we introduce {\\it probabilistic $p$-frames} that generalize the concepts of frames and $p$-frames. We characterize the minimizers of {\\it probabilistic $p$-frame potentials} in terms of probabilistic $p$-frames. The latter problem is solved completely for $0<p\\leq 2$, and for all even integers $p$. In particular, these last results generalize \\cite{Seidel:2001aa} as well as the recently introduced notion of the probabilistic frame potential in \\cite{Ehler:2010aa}\n\n\\smallskip \n\n\\textbf{Further relations to statistics:}\nBesides the results on FUNTFs used in \\cite{KentTaylor1994,Tyler:1987, Tyler:1988}, and mentioned above, frame theory has essentially evolved independently of statistical fields such as statistical shape analysis \\cite{Dryden:1998aa} and directional statistics \\cite{Mardia:2008aa}. Nevertheless, there still exist several overlaps, and to the best of our knowledge, these overlaps have not yet been fully explored. Recently, frame theory has been used in directional statistics \\cite{Ehler:2010ac}, where FUNTFs are utilized to investigate on statistical tests for directional uniformity and to model and analyze patterns found in granular rod experiments. We must point out that similar results were obtained earlier by Tyler in  \\cite{Tyler:1988}. \n\nProbabilistic tight frames are multivariate probability distributions whose second moments' matrix is a multiple of the identity, and they are used in \\cite{Ehler:2010aa} to obtain approximate FUNTFs. The latter approximation procedure is connected to a classical problem in multivariate statistics, namely estimating the population covariance from a sample, which is closely related to the $M$-estimators addressed in \\cite{KentTaylor1994,Tyler:1987,Tyler:1988}. The $p$-frame potentials and their probabilistic counterparts that we consider in the sequel, are linked to the notion of shape measure, shape space, and mean shape used in statistical shape analysis. In Section \\ref{label:section p frame potential}, we establish a precise connection between the full Procrustes estimate of mean shape \\cite[Definition 3.3]{Kent:1994aa} which is the eigenvector corresponding to the largest eigenvalue of the frame operator. Moreover, this eigenvalue coincides with the upper frame redundancy as introduced in \\cite{Bodmann:2010aa}. The full Procrustes estimate of mean shape also saturates the upper frame inequality. Moreover, the $p$-frame potentials form size measures as required in statistical shape analysis, and their minimizers among all collections of $N$ points on the sphere define a shape space modulo rotations. \n\nWe hope that the present paper will renew interests in more investigation on the role of frames and the $p$-th frame potential in directional statistics and statistical shape analysis. \n\n\n\n \n \\section{The $p$-frame potential}\\label{section:pfp}\n\n\\subsection{Background on frames and the frame potential}\nTo introduce frames and their elementary properties, we follow the textbook \\cite{Christensen:2003aa}.\n\\begin{definition}\\label{framedef}\nA collection of points $\\{x_i\\}_{i=1}^N\\subset\\R^d$ is called a \\emph{finite frame for $\\R^d$} if there are two constants $0<A\\leq B$ such that\n\\begin{equation}\\label{frameineq}\nA\\|x\\|^2 \\leq \\sum_{i=1}^N |\\langle x,x_i\\rangle|^2 \\leq B\\|x\\|^2,\\quad\\text{for all $x\\in\\R^d$.}\n\\end{equation}\nIf the frame bounds $A$ and $B$ are equal, the frame $\\{x_i\\}_{i=1}^N\\subset\\R^d$ is called \\emph{a finite tight frame for $\\R^d$}. In this case, \n\\begin{equation}\\label{deftightf}\nA\\|x\\|^2 = \\sum_{i=1}^N |\\langle x,x_i\\rangle|^2,\\quad\\text{for all $x\\in\\R^d$.}\n\\end{equation}\nA finite tight frame $\\{x_i\\}_{i=1}^N\\subset \\R^d$  consisting of unit norm vectors is  called a \\emph{finite unit norm tight frame (FUNTF)} for $\\R^d$. In this case, the frame bound is $A=N\/d$. \n\nA collection of unit vectors $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ is called \\emph{equiangular} if there exists a nonnegative constant $C$ such that  $|\\langle x_i,x_j\\rangle|=C$, for $i\\neq j$. \n\\end{definition}\n\n\n\nGiven a collection of $N$ points $\\{x_i\\}_{i=1}^N$ in $\\R^d$, the \\emph{analysis operator} is the mapping \n \\begin{equation*}\n F: \\R^d \\rightarrow \\R^N, \\quad x\\mapsto \\big(\\langle x,x_i \\rangle\\big)_{i=1}^N.\n \\end{equation*} \n Its adjoint operator is called the \\emph{synthesis operator} and given by \n \\begin{equation*}\n  F^*: \\R^N \\rightarrow\\R^d , \\quad (c_i)_{i=1}^N\\mapsto \\sum_{i=1}^N c_i x_i.\n \\end{equation*}\nUsing these operators, it is easy to see that $\\{x_i\\}_{i=1}^N$ is a frame if and only if the \\emph{frame operator} defined by\n\\begin{equation*}\nS:= F^* F : \\R^d \\rightarrow \\R^d,\\quad x\\mapsto \\sum_{i=1}^N \\langle x,x_i\\rangle x_i \n\\end{equation*}\nis positive, self-adjoint, and invertible. In this case, the following reconstruction formula holds\n\\begin{equation}\\label{eq:reconstruction for frames}\nx = \\sum_{j=1}^N  \\langle S^{-1} x_i,x\\rangle x_i,\\text{ for all $x\\in\\R^d$,}\n\\end{equation}\nand $\\{S^{-1} x_i\\}_{i=1}^N$, in fact, is a frame too, called the \\emph{canonical dual frame}. If $\\{x_i\\}_{i=1}^N$ is a frame, then $\\{S^{-1\/2}x_i\\}_{i=1}^N$ is a finite tight frame. Moreover, note that $\\{x_i\\}_{i=1}^N$ is a FUNTF if and only if its frame operator $S$ is $\\frac{N}{d}$ times the identity. \n\n\nAs mentioned in the introduction, the question of the existence and characterization of FUNTFs was settled in \\cite{Benedetto:2003aa}, where the \\emph{frame potential}, defined by \n\\begin{equation}\\label{eq:frame potential}\n\\FP(\\{x_i\\}_{i=1}^N) = \\sum_{i=1}^N\\sum_{j=1}^N |\\langle x_i,x_j\\rangle |^2,\n\\end{equation} was introduced and used to give a characterization of its minimizers in terms of FUNTFs. More specifically, they prove the following result: \n\n\n\\begin{theorem}\\cite[Theorem 7.1]{Benedetto:2003aa}\\label{theorem:Benedetto Fickus}\nLet $N$ be fixed and consider the minimization of the frame potential among all collections of $N$ points on the sphere $S^{d-1}$.\n\\begin{itemize}\n\\item[a)] If $N\\leq d$, then the minimum of the frame potential is $N$. The minimizers are exactly the orthonormal systems for $\\R^d$ with $N$ elements.\n\\item[b)] If $N\\geq d$, then the minimum of the frame potential is $\\frac{N^2}{d}$. The minimizers are exactly the FUNTFs for $\\R^d$ with $N$ elements.\n\\end{itemize}\n\\end{theorem}\n\nWe shall prove in the sequel that the frame potential is just an example in a family of functionals defined on points on the sphere, and whose minimizers have approximation properties similar to those of the frame potential. But first, we briefly comment on the relation between FUNTFs and $M$-estimators of multivariate scatter:\n\\begin{remark}\\label{remark:M estimator FUNTF}\nThe concept of FUNTFs is used in signal processing to represent a signal $x\\in\\R^d$ by means of $x=\\frac{d}{N}\\sum_{i=1}^N\\langle x,x_i\\rangle x_i$, which is similar to the well-known expansion in an orthonormal basis. FUNTFs have also been used in statistics to derive $M$-estimators of multivariate scatter: For a sample $\\{x_i\\}_{i=1}^N\\subset \\R^d$, where $x_i\\neq 0$, for $i=1\\ldots,N$, Tyler aims to find a symmetric positive definite matrix $\\Gamma$ such that \n\\begin{equation*\nM(\\Gamma) = \\frac{d}{N}\\sum_{i=1}^N \\frac{\\Gamma^{1\/2}x_i x_i' \\Gamma^{1\/2}}{x_i'\\Gamma x_i}\n\\end{equation*}\nis the identity matrix. Whenever this is possible, the estimate $V$ of the population scatter matrix is then given by $V=\\Gamma^{-1}$. We refer to  \\cite{Tyler:1987} for details. Note that $M(\\Gamma)=I_d$ implies that \n\\begin{equation*}\n\\bigg\\{\\frac{\\Gamma^{1\/2}x_i}{\\sqrt{x_i'\\Gamma x_i}}\\bigg\\}_{i=1}^{N}=\\bigg\\{\\frac{\\Gamma^{1\/2}x_i}{\\|\\Gamma^{1\/2}x_{i}\\|}\\bigg\\}_{i=1}^{N} \\subset S^{d-1}\n\\end{equation*}\n forms a FUNTF. Moreover, $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ is a FUNTF if and only if $M(I_d)=I_d$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\subsection{Definition of the $p$-frame potential}\\label{label:section p frame potential}\n\n\\begin{definition}\\label{pframepotential}\nLet $N$ be a positive integer, and $0<p< \\infty$. Given a collection of unit vectors $\\{x_i\\}_{i=1}^{N}\\subset S^{d-1}$, the \\emph{$p$-frame potential} is the functional\n  \\begin{equation}\\label{eq:pth frame potential}\n  \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})=\\sum_{i, j=1}^{N}|\\langle x_i,x_j\\rangle |^p. \n \\end{equation}\nWhen, $p=\\infty$, the definition reduces to\n\n\\begin{equation*}\n\\FP_{\\infty, N}(\\{x_{i}\\}_{i=1}^{N})=\\sup_{i\\neq j} |\\langle x_i,x_j\\rangle |.\n\\end{equation*}\n\\end{definition}\n\n\nIt is clear that the $p$-frame potential generalizes the frame potential in \\eqref{eq:frame potential}. Finite frames and the $p$-frame potential also extend to complex $\\{z_i\\}_{i=1}^N\\subset S_{\\C}^{d-1}=\\{z\\in\\C^d : \\|z\\|=1\\}$ and are related to statistical shape analysis, a tool to quantitatively track the physical deformation of objects. We refer to \\cite[Chapters 2, 3 $\\&$ 4]{Dryden:1998aa} for more details on shape analysis, but we briefly indicate here its link to the $p$-frame potential.  The shape of an object is specified by landmark points that altogether form the shape space. Often, a suitable transformation is applied first in order to study shape independently on the object's size. To remove the feature of size, we must specify a size measure $g$ (\\cite[Definition 2.2]{Dryden:1998aa}), which is a positive function defined on  $S_{\\C}^{d-1}$ that satisfies  \n\\begin{equation*}\ng(s\\{z_i\\}_{i=1}^N) = s g(\\{z_i\\}_{i=1}^N),\\quad\\text{for all $s\\in\\R_{+}$ and $\\{z_i\\}_{i=1}^N\\subset S_{\\C}^{d-1}$}.\n\\end{equation*}\nIt is immediate that the following family of functionals can be seen as size measures: \n\\begin{equation*}\ng_p(\\{z_i\\}_{i=1}^N) := \\big(\\sum_{i, j=1}^{N}|\\langle z_i,z_j\\rangle |^{p}\\big)^{\\frac{1}{p}} =\\bigg(\\FP_{N,p}(\\{z_i\\}_{i=1}^N)\\bigg)^{1\/p}.\n\\end{equation*}\nIn particular, $g_1$ represents the centroid size (when the shape is centered at the origin), which is one of the most common size measures in statistical shape analysis. Given two complex configurations $z_1,z_2\\in\\C^d$ derived from landmarks that code two-dimensional shape, the full Procrustes distance (\\cite[Definition 3.2]{Dryden:1998aa})  is \n\\begin{equation*}\nd_F(z_1,z_2)= \\inf_{\\beta,\\theta,a,b} \\bigg{\\|}\\frac{z_1}{\\|z_1\\|} - \\frac{z_2}{\\|z_2\\|}\\beta e^{i\\theta} -a -ib \\bigg{\\|}.\n\\end{equation*}\n Given $N$ configurations $\\{z_i\\}_{i=1}^N\\subset \\C^d$, the full Procrustes estimate of mean shape is defined by \n\\begin{equation*}\n\\bar{z}_P = \\arg\\inf_{\\|z\\|=1} \\big( \\sum_{i=1}^N d_F^2(z_i,z) \\big),\n\\end{equation*}\ncf.~\\cite[Definition 3.3]{Dryden:1998aa}. The average axis from the complex Bingham maximum likelihood estimator is the same as the full Procrustes estimate of mean shape for two-dimensional shapes, and the same holds for the complex Watson distribution, cf.~\\cite[Sections 6.2 $\\&$ 6.3]{Dryden:1998aa}. If we assume that $\\{z_i\\}_{i=1}^N$ are centered around zero, then $\\bar{z}_P$ is given by the eigenvector corresponding to the largest eigenvalue $\\lambda$ of the frame operator of the normalized collection $\\{\\frac{z_i}{\\|z_i\\|}\\}_{i=1}^N$. Furthermore, one observes that this eigenvalue $\\lambda$ is the upper frame redundancy of $\\{z_i\\}_{i=1}^N$ as introduced in \\cite{Bodmann:2010aa}, which also coincides with the optimal upper frame bound $B$ in \\eqref{frameineq}. Therefore, the full Procrustes estimate of mean shape satisfies the upper frame inequality with equality. Moreover, we have observed that the root mean square of the full Procrustes estimate of mean shape is $1-\\frac{\\lambda}{N}$.\n\n\n\n\n\nBefore stating the next elementary result, we recall some basic definitions in physics: A {\\it conservative force} $F$, is a vector field defined on $\\R^d$, such that $-F$ is the gradient of some {\\it potential} $P$ that is then induced by the conservative force. Lemma~\\ref{conservative} below was proved for the frame potential in \\cite{Benedetto:2003aa}, and we extend it to all $p$-frame potentials with $1<p<\\infty$:\n\\begin{lemma}\\label{conservative}\nFor each $p\\in (1, \\infty)$, the $p$-frame potential $\\FP_{p, N}$ is induced by the conservative force $F_p= f_p(\\|a-b\\|)(a-b)$, for $a,b\\in S^{d-1}$, where\n\\begin{equation*}\n f_p(x):=\\begin{cases}\n p(1-\\frac{x^2}{2})^{p-1},&\\text{for } |x|\\leq \\sqrt{2},\\\\\n -p(\\frac{x^2}{2}-1)^{p-1},&\\text{otherwise.}\n \\end{cases}\n \\end{equation*}\n\\end{lemma}\n$F_p$ is a central force between the `particles' $a$ and $b$ that we call the \\emph{$p$-frame force}.\n\\begin{proof}   The function  \n \\begin{equation*}\n \\mathbf{p}_p(x):=|1-\\frac{x^2}{2}|^p=\\begin{cases}\n (1-\\frac{x^2}{2})^p,&\\text{for } |x|\\leq \\sqrt{2},\\\\\n (\\frac{x^2}{2}-1)^p,&\\text{otherwise,}\n \\end{cases}\n \\end{equation*}\nis differentiable and satisfies  $ \\mathbf{p}_p'(x)=-xf_p(x)$. This is sufficient to verify that the potential $P_p(a,b):=\\mathbf{p}_p(\\|a-b\\|)$, defined for $a,b\\in S^{d-1}$, satisfies $\\nabla_a P_p(a,b) = -F(a,b)$, where $b$ is held fixed. Thus, $F_p$ is a conservative vector field. The physical meaningful potential $P_p(a,b)$ is in fact given by \n$$ P_p(a,b)  =\\mathbf{p}_p(\\|a-b\\|) = \\big |1-\\frac{1}{2}\\|a-b\\|^2 \\big |^p = \\big|\\langle a,b\\rangle\\big|^p,$$\n where we used that $\\|a\\|=\\|b\\|=1$. \n Consequently, the $p$-frame potential is induced by the conservative central force $F_p$. \n \\end{proof}\n \n As a consequence of the above lemma, $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ are in \\emph{equilibrium} under the $p$-frame force if they minimize the $p$-frame potential among all collections of $N$ points on the sphere. Note that such a collection of equilibria modulo rotations form a shape space. \n\n\\begin{remark}\nWe will use repeatedly the fact that for a fixed $\\{x_{i}\\}_{i=1}^{N} \\subset S^{d-1}$, the $p$-frame potential $\\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N})$ is a decreasing and continuous  function of $p \\in (0, \\infty)$. \n\n\\end{remark}\n\n\n \n\\section{Lower estimates for the $p$-frame potential}\\label{section:estimates}\nWe start this section with a few elementary results about the minimizers of the $p$-frame potential as well as their connection to $t$-designs. In fact, potentials on the sphere, and $t$-designs have been well investigated \\cite{Bachoc:2005aa, cokum06, Delsarte:1977aa, Seidel:2001aa, Venkov:2001aa}. However, one of the key differences between $t$-designs and our $p$-frame potential is that the former is considered only for positive integers $t$ while the latter is investigated for $p\\in (0, \\infty)$.  \n\n\\subsection{The Welch bound revisited}\\label{subsection:Welch}\nIf $p=2k$ is an even integer, one can use Welch's results~\\cite{Welch:1974aa} to conclude that, for $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$,\n\\begin{equation}\\label{eq:Welch}\n\\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}) \\geq \\frac{N^2}{\\binom{d+k-1}{k}}.\n\\end{equation}\nWe shall verify that this estimate is not optimal for small $N$, by proving an  estimate for $\\FP_{p, N}$ when $2<p<\\infty$. The following Proposition first appeared in \\cite{Oktay:2007}: \n\\begin{proposition}\\label{prop:potential for p>2}\nLet $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, $N\\geq d$, and $2<p<\\infty$, then\n\\begin{equation}\\label{eq:potential for p>2}\n\\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}) \\geq N(N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^{p\/2}+N,\n\\end{equation}\n and equality holds if and only if  $\\{x_i\\}_{i=1}^N$ is an equiangular FUNTF.\n\\end{proposition}\n\n\n\\begin{proof}\nFor $\\frac{1}{2}=\\frac{1}{p}+\\frac{1}{r}$, H\\\"older's inequality yields\n\\begin{equation}\\label{eq:hoelder numer 1}\n\\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_2} \\leq \\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_p}(N(N-1))^{1\/r}.\n\\end{equation}\nRaising to the $p$-th power and applying $\\frac{1}{r}=\\frac{1}{2}-\\frac{1}{p}$ leads to \n\\begin{equation}\\label{eq:raising to the power}\n\\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_2}^p \\leq \\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_p}^p(N(N-1))^{p\/2-1}.\n\\end{equation}\nTherefore, \n$$\n \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^p  \\geq  \\big(\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^2\\big)^{p\/2} (N(N-1))^{1-p\/2}.$$\nUsing the fact that $\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^2\\geq \\frac{N^2}{d}-N$ (see Theorem~\\ref{theorem:Benedetto Fickus}) implies that\n$$ \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^p \\geq \\big(N(\\frac{N}{d}-1)\\big)^{p\/2} (N(N-1))^{1-p\/2} = N(N-1)\\bigg(\\frac{N-d}{d(N-1)}\\bigg)^{p\/2},$$\nwhich proves~\\eqref{eq:potential for p>2}. \n\nTo establish the last part of the Proposition, we recall that an equiangular FUNTF $\\{x_{k}\\}_{k=1}^{N} \\subset \\R^d$ satisfies \n\\begin{equation}\\label{eq:equi}\n|\\langle x_i,x_j\\rangle | = \\sqrt{\\frac{N-d}{d(N-1)}},\\quad \\text{ for all }i\\neq j\n\\end{equation} \nsee, \\cite{Casazza:2008ab,Sustik:2007aa}, for details. Consequently, if $\\{x_{k}\\}_{k=1}^{N}$ is an equiangular FUNTF, then~\\eqref{eq:potential for p>2} holds with equality. \n\nOn the other hand, if equality holds in \\eqref{eq:potential for p>2}, then $\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^2 = \\frac{N^2}{d}-N$ and $\\{x_i\\}_{i=1}^N$ is a FUNTF due to Theorem \\ref{theorem:Benedetto Fickus}. Moreover, the H\\\"older estimate \\eqref{eq:hoelder numer 1} must have been an equality which means that $|\\langle x_i,x_j\\rangle|=C$ for $i\\neq j$, and some constant $C\\geq 0$. Thus, the FUNTF must be equiangular.\n\\end{proof}\n\n\n By comparing \\eqref{eq:Welch} with \\eqref{eq:potential for p>2}, it is easily seen that the Welch bound is not optimal for small $N$: \n \n \n \\begin{proposition}\\label{prop:second one}\n Let $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ and $p=2k>2$ be an even integer. If $d<N \\leq  \\binom{d+k-1}{k}$, then \n  \\begin{equation}\\label{eq:abc}\n \\FP_{p,N}(\\{x_{k}\\}_{k=1}^{N}) \\geq N(N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^k +N >\\frac{N^2}{\\binom{d+k-1}{k}}.\n \\end{equation} \n \\end{proposition}\n \n \n \\begin{proof} \n The condition on $N$ implies $1 \\geq   \\frac{N}{\\binom{d+k-1}{k}}$, and adding $ (N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^k>0$ to the right hand side leads to\n \\begin{equation*}\n  (N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^k +1 >  \\frac{N}{\\binom{d+k-1}{k}}.\n \\end{equation*}\n Multiplication by $N$ and Proposition \\ref{prop:potential for p>2} then yield \\eqref{eq:abc}.\n \\end{proof}\n\n\\begin{remark} The estimate in Proposition \\ref{prop:potential for p>2} is sharp if and only if an equiangular FUNTF exists. In \\cite[Sections 4 $\\&$ 6]{Sustik:2007aa}, construction (and hence existence) of equiangular FUNTFs was established when $d+2 \\leq N \\leq 100$. For general $d$ and $N$, a necessary condition for existence of equiangular FUNTFs is given, and it is conjectured that the conditions are sufficient as well. The authors essentially provide on upper bound on $N$ that depends on the dimension $d$. \nTherefore, Proposition~\\ref{prop:potential for p>2} might not be optimal when the redundancy $N\/d$ is much larger than $1$. \n\\end{remark}\n\n\n\\subsection{Relations to spherical $t$-designs}\\label{subsection:t design}\nA \\emph{spherical $t$-design} is a finite subset $\\{x_i\\}_{i=1}^N$ of the unit sphere $S^{d-1}$ in $\\R^d$,\nsuch that,\n\\begin{equation*}\n\\frac{1}{N}\\sum_{i=1}^N h(x_i) = \\int_{S^{d-1}} h(x)d\\sigma(x),\n\\end{equation*}\nfor all homogeneous polynomials $h$ of total degree equals or less than $t$ in $d$ variables and where $\\sigma$ denotes the uniform surface measure on $S^{d-1}$ normalized to have mass one. The following result is due to \\cite[Theorem 8.1]{Venkov:2001aa} (see \\cite{Seidel:2001aa}, \\cite{Delsarte:1977aa} for similar results). \n\n \n \\begin{theorem}\\label{theorem:p even integer discrete}\\cite[Theorem 8.1]{Venkov:2001aa}\nLet $p=2k$ be an even integer and $\\{x_i\\}_{i=1}^N=\\{-x_i\\}_{i=1}^N\\subset S^{d-1}$, then \n  \\begin{equation*}\n\\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})  \\geq  \\frac{1\\cdot 3\\cdot 5\\cdots(p-1)}{d(d+2)\\cdots (d+p-2) }N^2,\n \\end{equation*}\nand equality holds if and only if $\\{x_i\\}_{i=1}^N$ is a spherical $p$-design. \n \\end{theorem} \n \n \n \n\\subsection{Optimal configurations for  the $p$-frame potential}  \nWe first use Theorem~\\ref{theorem:Benedetto Fickus} to characterize the minimizers of the $p$-frame potential for $0<p<2$ provided that the number of points $N$ is a multiple of the dimension $d$:\n\n\n\\begin{theorem}\\label{prop:k multiples and small p}\nLet $0<p<2$ and assume that $N=kd$ for some positive integer $k$. Then the minimizers of the $p$-frame potential are exactly the $k$ copies of any orthonormal basis modulo multiplications by $\\pm 1$. The minimum of~\\eqref{eq:pth frame potential}, over all sets of $N=kd$ unit norm vectors, is $k^{2}d$.\n\\end{theorem}\n\\begin{proof}\nIf we fix a collection of vectors $\\{x_i\\}_{i=1}^N$, then the frame potential is a decreasing function in $p\\in(0, 2)$. Therefore, $$N^{2}\/d=\\FP_{2,N}(\\{x_{i}\\}_{i=1}^{N}) \\leq \\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N}).$$\n\nNow suppose that $\\{y_{i}\\}_{i=1}^{N}$ consists of $k$ copies of an  orthonormal basis $\\{e_{i}\\}_{i=1}^{d}$for $\\R^d$. Then,  $$\\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}) \\leq \\FP_{p, N}(\\{y_{i}\\}_{i=1}^{N})=\\FP(\\{y_{i}\\}_{i=1}^{N})=N^{2}\/d.$$ \nConsequently, $$\\min \\FP_{p, N}(\\{x_{k}\\}_{k=1}^{N})=\\min \\FP(\\{x_{k}\\}_{k=1}^{N})=N^{2}\/d.$$ \nThus, $\\FP_{p,N}$ has the same minimum as $\\FP$. Clearly $k$ copies of an orthonormal basis of $\\R^d$ form a  FUNTF, and hence minimize the $\\FP$ due to Theorem \\ref{theorem:Benedetto Fickus}. Consequently, the $k$ copies of an orthonormal basis also minimize the $p$-frame potential  for $0<p<2$. On the other hand, the inner products must be $0$ or $1$ to obtain a minimizer. The smallest $p$-frame potential then have the $k$ copies of an orthonormal basis.\n\\end{proof}\n\nWe now consider the $p$-frame potential for $N=d+1$. The case $p=2$ is covered by Theorem \\ref{theorem:Benedetto Fickus}. Note that any FUNTF with $N=d+1$ vectors is equiangular~\\cite{Goyal:2001aa,Strohmer:2003aa}. Hence, the case $2<p<\\infty$ is settled by Proposition~\\ref{prop:potential for p>2},  so we focus on $p\\in (0,2)$. \n\nOne easily verifies that, for $p_0=\\frac{\\log(\\frac{d(d+1)}{2})}{\\log(d)}$, an orthonormal basis plus one repeated vector and an equiangular FUNTF have the same $p_0$-frame potential $\\FP_{p_{0}, d+1}$. Under the assumption that those two systems are exactly the minimizers of $\\FP_{p_{0}, d+1}$, the next result will give a complete characterization of the minimizers of $\\FP_{p, d+1}$, for $0<p<2$. However, we have only been able to establish the validity of this assumption when $d=2$, cf.~Corollary \\ref{theorem:3 points}.\n\\begin{theorem}\\label{theorem:d+1}\nLet $N=d+1$ and set $p_{0}=\\frac{\\log(\\frac{d(d+1)}{2})}{\\log(d)}= \\frac{\\log(\\frac{N(N-1)}{2})}{\\log(N-1)}$. Let $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, and assume that $\\FP_{p_{0},N}(\\{x_{i}\\}_{i=1}^{N}) \\geq N+2,$ with equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an orthonormal basis plus one repeated vector or an equiangular FUNTF. Then,\n\\begin{itemize}\n\\item[\\textnormal{(1)}] for $0<p<p_{0} $, then for any $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, we have  $\\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N}) \\geq N+2,$ and equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an orthonormal basis plus one repeated vector,\n\\item[\\textnormal{(2)}] for $p_{0}<p<2$, then for any $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, we have $\\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N}) \\geq 2^{\\frac{p}{p_{0}}}\\, (Nd)^{1-\\frac{p}{p_{0}}} + N,$ \nand equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an equiangular FUNTF.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nUnder the assumptions of the Theorem, let $0<p<p_0$, then \n\\begin{equation*}\n\\min \\FP_{p_{0}, N}(\\{x_{i}\\}_{i=1}^{N}) \\leq \\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}),\n\\end{equation*}\nand using $\\{y_{i}\\}_{i=1}^{N}$ consisting of an orthonormal basis of $\\R^d$ with one repeated vector, yields that \n\\begin{equation*}\n\\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})\\leq \\FP_{p, N}(\\{y_{i}\\}_{i=1}^{N})=N+2=\\min \\FP_{p_{0}, N}(\\{x_{i}\\}_{i=1}^{N}). \n\\end{equation*}\nConsequently, $\\min \\FP_{p_{0}, N}(\\{x_{i}\\}_{i=1}^{N}) = \\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})=N+2$. Since an orthonormal basis plus one repeated vector minimizes the $p$-frame potential for $p=p_{0}$, it must also minimize $\\FP_{p, N}$ for  $0<p<p_{0}$, which proves $(1)$.\n\nAssume now that $p_0 < p < 2$. Choose $r$ such that $\\frac{1}{p_{0}}=\\frac{1}{p}+\\frac{1}{r}$, H\\\"older's  inequality yields\n\\begin{equation*}\n\\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_{p_{0}}}^p \\leq \\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_p}^p(N^{2}-N)^{p\/r}.\n\\end{equation*}\nSince we assume that $(2)$ holds, we have \n$ \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^{p_{0}} \\geq  2 $, which leads to \n\\begin{align*}\n \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^p & \\geq  \\bigg(\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^{p_{0}}\\bigg){\\frac{p}{p_{0}}} (N^{2}-N)^{1-\\frac{p}{p_{0}}}\\\\\n &\\geq 2^{\\frac{p}{p_{0}}}\\, (Nd)^{1-\\tfrac{p}{p_{0}}}.\n\\end{align*}\nThis concludes the proof of $(2)$. \nBy applying \\eqref{eq:equi}, one then checks that an equiangular FUNTF satisfies $(2)$ with equality.\n\n\nThe ``only'' part comes from the fact that the H\\\"older inequality becomes an equality  only if the sequences are linearly dependent. This means that the $\\{x_i\\}_{i=1}^N$ are equiangular. They must then satisfy $|\\langle x_i,x_j\\rangle | = \\frac{1}{d}$. Thus, by~\\eqref{eq:equi}  they form an equiangular FUNTF \\cite{Casazza:2008ab,Sustik:2007aa}.\n\\end{proof}\n\nWhen $d=2$ we can in fact verify the main hypothesis of Theorem~\\ref{theorem:d+1}, which leads to the following result: \n\n\n\\begin{corollary}\\label{theorem:3 points}\nLet $\\{x_i\\}_{i=1}^3\\subset S^{1}$, and set $p_{0}=\\frac{\\log(3)}{\\log(2)}$. Then, $$\\FP_{p_{0},3}(\\{x_{i}\\}_{i=1}^{3}) \\geq 5,$$ and equality holds if and only if $\\{x_i\\}_{i=1}^3$ is an orthonormal basis plus one repeated vector or an equiangular FUNTF. \n\nConsequently,\n\\begin{itemize}\n\\item[\\textnormal{(1)}] for $0<p<p_{0} $, then for any $\\{x_i\\}_{i=1}^3\\subset S^{1}$, we have $\\FP_{p,3}(\\{x_{i}\\}_{i=1}^{3}) \\geq 5,$ and equality holds if and only if $\\{x_i\\}_{i=1}^3$ is an orthonormal basis plus one repeated vector,\n\\item[\\textnormal{(2)}] for $p_{0}<p<\\infty$, then for any $\\{x_i\\}_{i=1}^3\\subset S^{1}$, we have $\\FP_{p,3}(\\{x_{i}\\}_{i=1}^{3}) \\geq \\frac{6}{2^{p}} +3,$ \nand equality holds if and only if $\\{x_i\\}_{i=1}^3$ is an equiangular FUNTF.\n\\end{itemize}\n\\end{corollary}\nThe minimum of $\\FP_{p,3}$, for $0<p<\\infty$ is plotted in Figure \\ref{figure:min of 2d 3 vectors}. \n \\begin{figure}\n \\centering\n  \\includegraphics[width=.35\\textwidth]{p_plot_2_b.eps}\\hspace{1ex}\n \\caption{Minimum of $\\FP_{p,3}$ in Corollary \\ref{theorem:3 points}, for $0<p<10$.}\\label{figure:min of 2d 3 vectors}\n \\end{figure}\n\\begin{proof}\nClearly  $(1)$ and $(2)$ follows from Theorem~\\ref{theorem:d+1} once the minimizers of $\\FP_{p_{0}, N}$ are characterized.  \n\nWithout loss of generality, let $\\beta$ be the smallest angle between $x_1$, $x_2$, and $x_3$ and let $\\alpha$ be the second smallest angle between them. This yields, of course, $0\\leq \\beta\\leq \\alpha$.\n\nCase 1: For $0\\leq  \\alpha+\\beta\\leq\\frac{\\pi}{2}$, we have \n\\begin{equation*}\n\\frac{1}{2}\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle|^p = \\cos^p(\\alpha) +\\cos^p(\\beta)+\\cos^p(\\alpha+\\beta)=F(\\alpha, \\beta).\n\\end{equation*} \nSince $1<p_{0}$, the $p$-frame potential is differentiable in $\\alpha$ and $\\beta$, and its critical points are \n\\begin{align*}\n0 & =-p\\cos^{p-1}(\\alpha)\\sin(\\alpha)-p\\cos^{p-1}(\\alpha+\\beta)\\sin(\\alpha+\\beta)\\\\\n0 & = -p\\cos^{p-1}(\\beta)\\sin(\\beta)-p\\cos^{p-1}(\\alpha+\\beta)\\sin(\\alpha+\\beta).\n\\end{align*}\nThis implies that either $\\alpha=\\beta=0$ or $\\beta=0$ and $\\alpha=\\frac{\\pi}{2}$. In the first case, we have a maximum since it implies $x_1=x_2=x_3$. The latter case means that two points are identical and the third one is perpendicular which is a potential minimum of the $p$-frame potential. \n\nCase 2: We can assume that $\\frac{\\pi}{4}\\leq \\alpha$ (otherwise we are in Case 1). We can further assume that $\\frac{\\pi}{4}\\leq \\alpha \\leq \\frac{\\pi}{2}\\leq \\alpha+\\beta\\leq \\frac{2\\pi}{3}$ (if $\\frac{2\\pi}{3}<\\alpha+\\beta\\leq\\pi$. Otherwise, substitute $x_i$ with $-x_i$.\nWe now have \n\\begin{equation*}\n\\frac{1}{2}\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle|^p =\\cos^p(\\alpha) +\\cos^p(\\beta)+(-\\cos(\\alpha+\\beta))^p=G(\\alpha, \\beta).\n\\end{equation*} \nThe critical points of $G$ are given by\n\\begin{align*}\n0 & =-p\\cos^{p-1}(\\alpha)\\sin(\\alpha)+p(-\\cos(\\alpha+\\beta))^{p-1}\\sin(\\alpha+\\beta)\\\\\n0 & = -p\\cos^{p-1}(\\beta)\\sin(\\beta)+p(-\\cos(\\alpha+\\beta))^{p-1}\\sin(\\alpha+\\beta).\n\\end{align*}\nBy subtracting one equation from the other and raising to the second power, we obtain\n \\begin{equation*}\n z_1^{p-1}(1-z_1) = z_2^{p-1}(1-z_2),\n \\end{equation*}\n where $z_1=\\cos(\\alpha)^2$ and $z_2=\\cos^2(\\alpha+\\beta)$. Since $\\sin^2(x)-1\/2\\geq \\cos^2(x)$ for all $\\pi\/2\\leq x\\leq 2\\pi\/3$, we have $0\\leq z_1\\leq 1\/2$ and $0\\leq z_2\\leq 1\/4$ because $z_2\\leq 1-z_2-1\/2$.\n \n We consider the function $F(z)=z^{p-1}(1-z)$ on $0\\leq z\\leq 1\/2$. $F$ achieves its maximum at $z=\\frac{p-1}{p}\\approx 0.3691$ and is convex, cf.~Figure \\ref{figure:1}.%\n \\begin{figure}\n \\centering\n \\subfigure[$F(z)=z^{p-1}(z-1)$, $0\\leq z\\leq \\frac{1}{2}$]{\\hspace{1ex}\n \\includegraphics[width=.35\\textwidth]{function_1.eps}\\hspace{1ex}}\n \\hspace{2ex}\n \\subfigure[$f(\\alpha) =  2\\cos^p(\\alpha) + (-\\cos(2\\alpha))^p$, $\\frac{\\pi}{3}\\leq \\alpha\\leq \\frac{\\pi}{2}$]{\\hspace{7ex}\n \\includegraphics[width=.35\\textwidth]{function_3.eps}\\hspace{7ex}}\n \\caption{$F$ and $f$ in the proof of Theorem \\ref{theorem:3 points}}\\label{figure:1}\n \\end{figure}\n Therefore, for $0\\leq z_1\\leq 1\/2$ and $0\\leq z_2\\leq 1\/4$, we have $F(z_1)=F(z_2)$ if and only if $z_1=z_2$ or $z_1=1\/2$ and $z_2=1\/4$. For $z_1=\\cos^2(\\alpha)=1\/2$, we have $\\alpha=\\pi\/3$ and $z_2=\\cos^2(\\pi\/3+\\beta)=1\/4$ yields $\\beta=\\pi\/3$. The case $z_1=z_2$ leads to $\\cos^2(\\alpha)=\\cos^2(\\alpha+\\beta)$ which implies $\\alpha+\\beta-\\pi\/2 = \\pi\/2-\\alpha$. This is equivalent to $\\beta=\\pi-2\\alpha$. Since we assume $\\beta\\leq \\alpha$, we obtain $\\pi\/3\\leq \\alpha \\leq \\pi\/2$. \n\nWe now check the minima of the function\n \\begin{align*\nf(\\alpha)  & =  \\cos^p(\\alpha)+\\cos^p(\\pi-2\\alpha)+(-\\cos(\\pi-\\alpha))^p\\\\\n& = 2\\cos^p(\\alpha) + (-\\cos(2\\alpha))^p,\n \\end{align*}\n on $\\pi\/3\\leq\\alpha\\leq \\pi\/2$, cf.~Figure \\ref{figure:1}. Its derivative \n \\begin{equation*}\n \\frac{\\partial f}{\\partial \\alpha} (\\alpha) = -2p\\cos^{p-1}(\\alpha)\\sin(\\alpha) + 2p(-\\cos(2\\alpha))^{p-1}\\sin(2\\alpha)\n \\end{equation*}\nvanishes if and only if \n \\begin{equation*}\n \\cos^{p-1}(\\alpha)\\sin(\\alpha) = (\\sin^2(\\alpha)-\\cos^2(\\alpha))^{p-1}2\\cos(\\alpha)\\sin(\\alpha)\n \\end{equation*}\n For $\\alpha\\neq \\pi\/2$, this yields\n  \\begin{equation*}\n \\cos^{p-1}(\\alpha) = (1-2\\cos^2(\\alpha))^{p-1}2\\cos(\\alpha).\n \\end{equation*}\n By substituting $x=\\cos(\\alpha)$, we obtain, for $0<x\\leq 1\/2$\n  \\begin{equation*}\n x^{p-1} = (1-2x^2)^{p-1}2x,\n \\end{equation*}\nwhich is equivalent to \n  \\begin{equation*}\n 1\/2 = (1\/x-2x)^{p-1}x \\Leftrightarrow\n 0 = (1\/x-2x)x^{1\/(p-1)} -(1\/2)^{1\/(p-1)}\n \\end{equation*}\nand define the new function \n   \\begin{equation*}\n g(x) = -2x^{q+1}+x^{q-1} -(1\/2)^{q},\n \\end{equation*}\n where $q=1\/(p-1)$. To show that $g$ has only one extremal point on $0<x\\leq 1\/2$, we differentiate\n   \\begin{equation*}\n  \\frac{\\partial g}{\\partial x}(x) = -2(q+1)x^q+(q-1)x^{q-2}.\n \\end{equation*}\n The term $ \\frac{\\partial g}{\\partial x}(x)$ vanishes if and only if \n \\begin{equation*}\n (q-1)x^{q-2} = 2(q+1)x^q,\n \\end{equation*}\nwhich is equivalent to\n \\begin{equation*}\n x^2 = \\frac{q-1}{2(q+1)} =\\frac{2-\\log_2(3)}{2\\log_2(3)}.\n \\end{equation*}\n Hence, $x\\approx 0.3618$ and $ \\frac{\\partial g}{\\partial x}$ does not have any other zeros on $0<x\\leq 1\/2$. This means that $g$ has only one extremal point and can then only have two zeros on $0<x\\leq 1\/2$. The zero at $x=1\/2$ corresponds to a minimum. This means that the zero of $ \\frac{\\partial g}{\\partial x}$ at $x\\approx 0.3618$ is a maximum of $g$. Hence the other zero of g is between $0$ and $\\approx 0.3618$. However, this other zero corresponds to a maximum of $f$. The minimum of $f$ can thus be at $x=0$ or $x=1\/2$. This implies $\\alpha=\\pi\/3$ or $\\alpha=\\pi\/2$. It is easy to verify that $\\alpha=\\pi\/3$ would lead to $\\beta=\\pi\/3$, and $\\alpha=\\pi\/2$ yields $\\beta=0$. Thus, the minimum of the $p$-frame potential corresponds to either an orthonormal basis plus one repeated element ($\\alpha=\\pi\/2$, $\\beta=0$) or an equiangular FUNTF ($\\alpha=\\beta=\\pi\/3$). One easily checks that both situations lead to the same global minimum.\n \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\nIn view of Theorem \\ref{theorem:d+1} and Corollary \\ref{theorem:3 points}, we have the following conjecture: \n\\begin{conjecture}\\label{conjecture:p_0}\nLet $N=d+1$  and $p_0=\\frac{\\log(\\frac{d(d+1)}{2})}{\\log(d)}$. Then $$\\FP_{p_{0},N}(\\{x_{k}\\}_{k=1}^{N}) \\geq N+2,$$ and equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an orthonormal basis plus one repeated vector or an equiangular FUNTF.\n\\end{conjecture}\n\nOne can check that $1<p_0<2$, for $d>1$. According to Proposition \\ref{prop:potential for p>2}, the minimizers of the $p$-frame potential for $2<p<\\infty$ are exactly the equiangular FUNTFs. Thus, our conjecture essentially addresses the range $0<p<2$. \n\n\\begin{remark}\nAlthough we are primarily interested in real unit norm vectors $\\{x_i\\}_{i=1}^N\\subset \\R^d$, we should mention that Theorem \\ref{theorem:Benedetto Fickus}, Theorem~\\ref{prop:k multiples and small p}, Equation \\eqref{eq:Welch}, and Propositions \\ref{prop:potential for p>2} and \\ref{prop:second one} still hold for complex vectors $\\{z_i\\}_{i=1}^N\\subset \\C^d$ that have unit norm. The constraints on $N$ and $d$ that allow for the existence of a complex FUNTF are slightly weaker than in the real case~\\cite{Sustik:2007aa}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n \n\\section{The probabilistic $p$-frame potential}\\label{section:intro prob}\n\nThe present section is dedicated to introducing a probabilistic version of the previous section. We shall consider probability distributions on the sphere rather than finite point sets. Let $\\mathcal{M}(S^{d-1},\\mathcal{B})$ denote the collection of probability distributions on the sphere with respect to the Borel sigma algebra $\\mathcal{B}$. \n\nWe begin by introducing the probabilistic $p$-frame which generalizes the notion of probabilistic frames introduced in~\\cite{Ehler:2010aa}. \n\n\\begin{definition}\\label{probpframe}\nFor $0<p<\\infty$, we call $\\mu\\in\\mathcal{M}(S^{d-1},\\mathcal{B})$ a \\emph{probabilistic $p$-frame} for $\\R^d$ if and only if there are constants $A,B>0$ such that\n \\begin{equation}\\label{ppframeineq}\nA\\|y\\|^p\\leq  \\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) \\leq B\\|y\\|^p, \\quad\\forall y\\in\\R^d.\n \\end{equation}\n We call $\\mu$ a \\emph{tight probabilistic $p$-frame} if and only if we can choose $A=B$.\n \\end{definition}\n Due to Cauchy-Schwartz, the upper bound $B$ always exists. \nConsequently, in order to check that $\\mu$ is a probabilistic $p$-frame one only needs to focus on the lower bound $A$. \n\n Since the uniform surface measure $\\sigma$ on $S^{d-1}$ is invariant under orthogonal transformations, one can easily check that it constitutes a tight probabilistic $p$-frame, for any $0<p<\\infty$. \n\n \n\n\nGiven a probability measure $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$, we call \n\\begin{equation*}\nF : \\R^d \\rightarrow L_p(S^{d-1},\\mu),\\quad x\\mapsto \\langle x,\\cdot\\rangle \n\\end{equation*}\nthe \\emph{analysis operator}. It is trivially seen that $$\\|F(y)\\|_{L_{p}(S^{d-1}, \\mu)}=\\|\\langle x,y\\rangle\\|_{L_{p}(S^{d-1}, \\mu)} \\leq \\|y\\|$$ for all $0<p\\leq \\infty$. The dual of $F$ is called \\emph{synthesis operator} and is given by\n\\begin{equation*}\nF^* : L_q(S^{d-1},\\mu) \\rightarrow \\R^d,\\quad f\\mapsto \\int_{S^{d-1}} f(x)xd\\mu(x),\n\\end{equation*}\nwhere $1=\\frac{1}{p}+\\frac{1}{q}$, and $1\\leq p \\leq \\infty$. In fact, $F^*$ is well-defined and bounded operator on all $L_{r}(S^{d-1}, \\mu)$ where \n$1\\leq r \\leq \\infty$. Indeed, for  $f \\in L_{r}(S^{d-1}, \\mu)$ we have \n $$\n\\|F^{*}(f)\\|\\leq \\int_{S^{d-1}}|f(x)|\\|x\\|d\\mu(x)= \\int_{S^{d-1}}|f(x)|d\\mu(x)\\leq \\|f\\|_{L_{r}}.$$ \nGiven $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$, the second moments matrix of $\\mu$ is the $d\\times d$ matrix defined by \n\\begin{equation}\\label{ppfoperator}\nS=F^* F = \\bigg(\\int_{S^{d-1}} x_i x_j d\\mu(x) \\bigg)_{i,j}\n\\end{equation}\nwith respect to the canonical basis for $\\R^d$. As we show next, the second moments matrix $S$ plays a key role in determining if $\\mu$ is a probabilistic $p$-frame. We refer to \\cite{Christensen:2003ab}, where a  result was proved for similar discrete $p$-frames. \n\n\n\\begin{proposition}\\label{pfonto}\nIf $1\\leq p <\\infty$, then $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$ is a probabilistic $p$-frame if and only if $F^*$ is onto. \n\\end{proposition}\n\n\\begin{proof}\nAs mentioned earlier the upper bound in the probabilistic $p$-frame definition always holds. So we only need to show the equivalence between the lower bound and the surjectivity of $F^{*}$. \n\nAssume that $F^{*}$ is surjective. Since $F$ is injective, then $S$ is invertible and for each $y \\in \\R^d$ we have\n$$\\|y\\|^{2}=|\\langle Sy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle F^{*}Fy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle Fy,FS^{-1}y\\rangle_{L^{p}\\to L^{p'}}|,$$ which can be estimated as follows:\n$$\\|y\\|^{2} \\leq \\|F(y)\\|_{L^{p}}\\|F(S^{-1} y)\\|_{L^{p'}}\\leq C \\|F(y)\\|_{L^{p}}\\|S^{-1} (y)\\|_{\\R^{d}}\\leq C \\|F(y)\\|_{L^{p}}\\|y\\|_{\\R^{d}}.$$\nTherefore, for all $y\\neq 0 \\in \\R^d$ we have $1\/C \\|y\\| \\leq \\|F(y)\\|_{L^{p}}$ that is $$ 1\/C \\|y\\|^{p}\\leq \\int_{S^{d-1}}|\\langle x,y\\rangle|^p d\\mu(x).$$ Thus $\\mu$ is a probabilistic $p$-frame.\n\nNow assume that $\\mu$ is a probabilistic $p$-frame, but that $F^{*}$ is not surjective. Then, there exists $z \\neq 0 \\in \\R^d$ such that $\\langle z, F^{*}f\\rangle=0$ for all $f \\in L_{p'}(S^{d-1}, \\mu)$. Consequently $$\\langle z, F^{*}f\\rangle_{\\R^{d}}=\\langle F(z), f\\rangle_{L_{p}\\to L_{p'}}=\\int_{S^{d-1}}f(x)\\langle x , z\\rangle, d\\mu(x)=0$$ for all $f \\in L_{p'}$. A contradiction argument leads to $\\langle x, z\\rangle =0$ for all $x \\in S^{d-1}$ which implies that $z=0$. Thus $F^{*}$ is surjective.\n\\end{proof}\n \nThe second moments matrix can also be  used to show that a probabilistic $p$-frame gives rise to a reconstruction formula that extends the finite frame expansion in \\eqref{eq:reconstruction for frames}. In addition, the next result generalizes the reconstruction formula for tight probabilistic frames obtained in \\cite[Lemma 3.7]{Ehler:2010aa}.\n\n\n\\begin{proposition}\\label{reconppf} Let $0<p<\\infty$ and assume that $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$ is a probabilistic $p$-frame for $\\R^d$. Set $\\tilde{\\mu}=\\mu \\circ S$. Then for each $y \\in \\R^d$, we have\n\\begin{equation}\\label{reconsppf}\ny= \\int_{S^{-1}(S^{d-1})}Sz \\, \\langle z, y\\rangle \\, d\\tilde{\\mu}(z)= \\int_{S^{-1}(S^{d-1})}z \\, \\langle Sz, y\\rangle \\, d\\tilde{\\mu}(z).\n\\end{equation} \n\\end{proposition}\n\n\\begin{proof}\nThe result follows by noticing that $y=SS^{-1}y=S^{-1}Sy$.\n\\end{proof}\n\n\nThe above result motivates the following definition:\n\n\\begin{definition}\\label{candualppf} Let $0< p < \\infty$. If $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$ is a probabilistic $p$-frame with frame operator $S$, then $\\tilde{\\mu}=\\mu \\circ S \\in \\mathcal{M}(S^{-1}(S^{d-1}),S^{-1}\\mathcal{B})$ is called the \\emph{probabilistic canonical dual frame} of $\\mu$\n\\end{definition}\n\nNote that if $\\mu$ in Proposition \\ref{reconppf} is the counting measure corresponding to a FUNTF $\\{x_i\\}_{i=1}^N$, then $\\tilde{\\mu}$ is the counting measure associated to the canonical dual frame of $\\{x_i\\}_{i=1}^N$. \n\n\n\n\n\n\n\n\\begin{lemma}\\label{equivappf} a) If $\\mu$ is probabilistic frame, then it is a probabilistic $p$-frame for all $1\\leq p< \\infty$. Conversely, if $\\mu$ is a probabilistic $p$-frame for some $1\\leq p < \\infty$, then it is a probabilistic frame.\n\n\\noindent b) Let $1\\leq p < \\infty$. If $\\mu$ is a probabilistic $p$-frame, then so is the canonical dual $\\tilde{\\mu}$.\n\\end{lemma}\n\n\n\\begin{proof}\na) \nAssume that $\\mu$ is a probabilistic frame and let $1\\leq p < \\infty$. Then, we only need to check that the lower inequality of~\\eqref{ppframeineq} holds, since the corresponding upper bound is trivial. \nBy Proposition~\\ref{pfonto} (applied to $p=2$), $S=F^{*}F$ is invertible and for each $y \\in \\R^d$ we have\n$$\\|y\\|^{2}=|\\langle Sy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle F^{*}Fy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle Fy,FS^{-1}y\\rangle_{L^{p}\\to L^{p'}}|,$$ which can be estimated as follows:\n$$\\|y\\|^{2} \\leq \\|F(y)\\|_{L_{p}}\\|F(S^{-1} y)\\|_{L_{p'}}\\leq C \\|F(y)\\|_{L_{p}}\\|S^{-1} (y)\\|_{\\R^{d}}\\leq C \\|F(y)\\|_{L_{p}}\\|y\\|_{\\R^{d}}.$$ Therefore, for all $y\\neq 0 \\in \\R^d$ we have $1\/C \\|y\\| \\leq \\|F(y)\\|_{L_{p}}$ that is $$ 1\/C \\|y\\|^{p}\\leq \\int_{S^{d-1}}|\\langle x,y\\rangle|^p d\\mu(x).$$ \nFor the converse, assume that $\\mu$ is a probabilistic $p$-frame for some $p>2$. Then, for all $y\\neq 0 \\in \\R^d$, \n\\begin{align*}\nA\\|y\\|^p&\\leq  \\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x)\\\\\n&=\\int_{S^{d-1}} |\\langle x,y\\rangle|^2\\, |\\langle x,y\\rangle|^{p-2}\\, d\\mu(x)\\\\\n&  \\leq \\int_{S^{d-1}} \\|x\\|^{p-2}\\, \\|y\\|^{p-2}\\, |\\langle x,y\\rangle|^2 d\\mu(x) \\\\\n&= \\|y\\|^{p-2}\\,  \\int_{S^{d-1}} |\\langle x,y\\rangle|^2 d\\mu(x),\n\\end{align*}from which it follows that $$A\\|y\\|^2 \\leq \\int_{S^{d-1}} |\\langle x,y\\rangle|^2 d\\mu(x).$$\n\nIf $\\mu$ is a probabilistic $p$-frame for some $p<2$. Then, for all $y\\neq 0 \\in \\R^d$, \n$$\\|y\\|^{2}=|\\langle Sy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle F^{*}Fy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle Fy,FS^{-1}y\\rangle_{L_{p}\\to L_{p'}}|,$$ which can be estimated by  \n$$\\|y\\|^{2} \\leq \\|Fy\\|_{L_{p}}\\|FS^{-1}y\\|_{L_{p'}}\\leq C \\|Fy\\|_{L_{2}}\\|y\\|,$$ where we have used the fact that for $p<2$, $L_{2}(S^{d-1}, \\mu) \\subset L_{p}(S^{d-1}, \\mu)$. This conclude the proof of a). \n\n\nb) If $\\mu$ is a probabilistic $p$-frame for some $1\\leq p < \\infty,$ then by a) $\\mu$ is a probabilistic frame. In this case, $\\tilde{\\mu}$ is known to be a probabilistic frame, cf.~\\cite{Ehler:2010aa}, and thus a probabilistic $p$-frame. \n\\end{proof}\n\n\n \n\n\n\nWe are particularly interested in tight probabilistic $p$-frame potentials, which we seek to characterize in terms of minimizers of appropriate potentials. This motivates the following definition: \n\n \\begin{definition}\\label{profframpot}\nFor $0<p<\\infty$ and $\\mu\\in\\mathcal{M}(S^{d-1},\\mathcal{B})$, the \\emph{probabilistic $p$-frame potential} is defined by  \n  \\begin{equation}\\label{eq:prob p pot}\n \\PFP(\\mu,p) = \\int_{S^{d-1}}\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y).\n \\end{equation}\n\\end{definition}\n\nFrom the weak-star-compactness of the collection of all probability distributions on the sphere, we can deduce that $\\PFP(\\mu, p)$ admits a minimizer which satisfies\n\\begin{equation}\\label{minppfp}\n\\PFP(p) = \\min_{\\mu\\in\\mathcal{M}(S^{d-1},\\mathcal{B})} \\PFP(\\mu,p).\n\\end{equation}\n\n \n\n \nWe now turn to the minimizers of the probabilistic frame potential $\\PFP(\\mu)$. In the process, we extend some ideas developed  in \\cite{Bjoerck:1955aa} to the probabilistic frame potential. \n\\begin{proposition}\\label{prop:1}\nLet $0<p<\\infty$ and let $\\mu$ be a minimizer of \\eqref{eq:prob p pot}, then \n\\begin{itemize}\n\\item[\\textnormal{(1)}] $\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) = \\PFP(p)$, for all $y\\in\\supp(\\mu)$,\n\\item[\\textnormal{(2)}] $\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x)\\geq \\PFP(p)$, for all $y\\in S^{d-1}$.\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\nThe proof will use the following observation. Let  $\\mu$ be a probability measure on $S^{d-1}$ and choose a measure  $\\nu$, such that $\\nu(S^{d-1})=0$ and $\\mu+\\varepsilon \\nu\\geq 0$, for all $0\\leq \\varepsilon\\leq 1$. Let us also introduce the notation $$ \\PFP(\\mu,\\nu,p):=\\int_{S^{d-1}}\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\nu(y).$$ We then obtain\n\\begin{align*}\n\\PFP(\\mu) & \\leq \\PFP(\\mu+\\varepsilon \\nu,p)\\\\\n& = \\PFP(\\mu,p)+\\varepsilon^2\\PFP(\\nu,p) + 2\\varepsilon \\PFP(\\mu,\\nu,p)\\\\\n& = \\PFP(\\mu)+\\varepsilon^2\\PFP(\\nu,p) + 2\\varepsilon \\PFP(\\mu,\\nu,p).\n\\end{align*} \nWe thus have $0\\leq \\varepsilon \\PFP(\\nu,p)+2\\PFP(\\mu,\\nu,p)$, for all $0\\leq \\varepsilon\\leq 1$, which implies $\\PFP(\\mu,\\nu,p)\\geq 0$. \n\nWe now prove $(1)$ using a contradiction argument. In particular, assume that $(1)$ does not hold. This implies that there are $y_1, y_2\\in \\supp(\\mu)$ such that \n\\begin{equation*}\na := \\int_{S^{d-1}} |\\langle x,y_2\\rangle|^p d\\mu(x) < \\int_{S^{d-1}} |\\langle x,y_1\\rangle|^p d\\mu(x) =: b.\n\\end{equation*}\nSet $P_\\mu(y)=\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x)$. \nLet $K$ be an open ball around $y_1$ in $S^{d-1}$ and so small that $y_2\\not\\in K$ and that the oscillation of $P_\\mu(y)$ on $K$ is smaller than $\\frac{b-a}{2}$. Let $m=\\mu(K)>0$. One can check that the measure $\\nu$ defined by \n\\begin{equation*}\n\\nu(E) := m\\delta_{y_2}(E)-\\mu(E\\cap K),\\quad E\\in\\mathcal{B},\n\\end{equation*}\nsatisfies $\\nu(S^{d-1})=0$, and $\\mu +\\epsilon \\nu \\geq 0$. Hence, $\\PFP(\\mu,\\nu,p)\\geq 0$. On the other hand, we can estimate\n$$\n\\PFP(\\mu,\\nu,p)  = \\int_{S^{d-1}} P_\\mu(y)d\\nu(y)= P_\\mu(y_2)m - \\int_K P_\\mu(y)d\\mu(y)= am - \\int_K P_\\mu(y)d\\mu(y)$$ and so\n\n$$\\PFP(\\mu,\\nu,p)  \\leq am - (b-\\frac{b-a}{2})m  = - \\frac{b-a}{2}m <0.$$\n\nThis is a contradiction to $\\PFP(\\mu,\\nu,p)\\geq 0$ and implies that there is a constant $C$ such that $P_\\mu(y)=C$, for all $y\\in\\supp(\\mu)$.  \nWe still have to verify that the constant $C$ is in fact $\\PFP(p)$: \n\\begin{align*}\n\\PFP(p)  = \\PFP(\\mu,p) & =  \\int_{S^{d-1}} P_\\mu(y)d\\mu(y) \\\\\n& =  \\int_{\\supp(\\mu)} P_\\mu(y)d\\mu(y)\\\\\n& =  \\int_{\\supp(\\mu)} C d\\mu(y) = C.\n\\end{align*}\n\nThe proof of $(2)$ is similar to the one above, and so we omit it. \n\\end{proof}\nThe following result is an immediate consequence of Proposition~\\ref{prop:1}. \n\n\\begin{corollary}\\label{theorem:tight p frame is necessary}\nLet $0<p<\\infty$ and let $\\mu$ be a minimizer of \\eqref{eq:prob p pot}, then\n\\begin{itemize}\n\\item[\\textnormal{(1)}] $\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) = \\PFP(p)\\|y\\|^p$, for all $\\frac{y}{\\|y\\|}\\in\\supp(\\mu)$,\n\\item[\\textnormal{(2)}] $\\supp(\\mu)$ is a complete subset of $\\R^d$. \n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\n$ (1)$ directly follows from $(1)$ in Proposition \\ref{prop:1}.\n\n$(2)$ If $\\supp(\\mu)$ is not complete in $\\R^d$, then there is an element $y\\in S^{d-1}$ that is in the orthogonal complement. However, this contradicts (2) in Proposition \\ref{prop:1}. \n\\end{proof}\n\nWe can now characterize the minimizers of the probabilistic $p$-frame potential when  $0<p<2$. In fact, we shall show that these minimizers are discrete probability measures, and the following theorem is the analogue of Proposition \\ref{prop:k multiples and small p}:\n \\begin{theorem}\\label{theorem:p less than 2}\n Let $0<p<2$, then the minimizers of  \\eqref{eq:prob p pot} are exactly those probability distributions $\\mu$ that satisfy both, \n \\begin{itemize}\n \\item[(i)] there is an orthonormal basis $\\{x_1,\\ldots,x_d\\}$ for $\\R^d$ such that \n \\begin{equation*}\n \\{x_1,\\ldots,x_d\\} \\subset \\supp(\\mu) \\subset \\{\\pm x_1,\\ldots,\\pm x_d\\},\n \\end{equation*}\n \\item[(ii)] there is $f:S^{d-1}\\rightarrow \\R$ such that $\\mu(x) = f(x) \\nu_{\\pm x_1,\\ldots,\\pm x_d}(x)$ and\n \\begin{equation*}\n f(x_i) +f(-x_i) = \\frac{1}{d}.\n \\end{equation*} \n \\end{itemize} \n \\end{theorem}\n The measure $\\nu_{\\pm x_1,\\ldots,\\pm x_d}(x)$ in Theorem \\ref{theorem:p less than 2} denotes the counting measure of the set $\\{\\pm x_i : i=1,\\ldots,d\\}$.\n \n \n \\begin{proof}\n Since $0\\leq |\\langle x,y\\rangle|\\leq 1$, for $x,y\\in S^{d-1}$, we have $\\PFP(\\mu,p)\\geq \\PFP(\\mu,2)$. In ~\\cite[Theorem 3.10]{Ehler:2010aa} it was shown that the normalized counting measure $\\frac{1}{d}\\nu_{x_1,\\ldots,x_d}$ of an orthonormal basis minimizes $\\PFP(\\cdot,2)$. Due to $\\PFP(\\frac{1}{d}\\nu_{x_1,\\ldots,x_d},2) = \\PFP(\\frac{1}{d}\\nu_{x_1,\\ldots,x_d},p)$, we obtain that $\\frac{1}{d}\\nu_{x_1,\\ldots,x_d}$ also minimizes $\\PFP(\\cdot,p)$ and hence $\\PFP(p)=\\PFP(2)$. \n \n In the following, we prove that all minimizers of $\\PFP(\\cdot,p)$ are essentially induced by an orthonormal basis. Let $\\mu$ be a minimizer and let $v,w\\in \\supp(\\mu)$. We first show that $|\\langle v,w\\rangle |\\in \\{0,1\\}$. The implications $\\langle v,w\\rangle =1$ if and only if $v=w$ and $\\langle v,w\\rangle =-1$ if and only if  $v=-w$ are trivial. \n \n Suppose now that $v\\neq \\pm w$ and $\\langle v,w\\rangle \\neq 0$, then there exist  $\\varepsilon>0$ and $\\delta_\\varepsilon>0$ such that \n \\begin{itemize}\n \\item[(a)] $B_\\varepsilon(v)\\cap B_\\varepsilon(w)=\\emptyset$ and $\\mu(B_\\varepsilon(v)), \\mu(B_\\varepsilon(w)) \\geq \\delta_\\varepsilon$. \n \\item[(b)] for all $x\\in B_\\varepsilon(v)$ and $y\\in B_\\varepsilon(w)$, $|\\langle x,y\\rangle |^p\\geq |\\langle x,y\\rangle |^2+\\varepsilon$.\n \\end{itemize}\nBy using $B=B_\\varepsilon(v)\\times B_\\varepsilon(w)$, this implies\n\\begin{align*}\n\\PFP(\\mu,p) & = \\int_{B} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y) + \\int_{S^{d-1}\\times S^{d-1}\\setminus B } |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y)\\\\\n& \\geq \\int_{B} (|\\langle x,y\\rangle|^2+\\varepsilon) d\\mu(x) d\\mu(y) + \\int_{S^{d-1}\\times S^{d-1}\\setminus B } |\\langle x,y\\rangle|^2 d\\mu(x) d\\mu(y)\\\\\n& = \\PFP(\\mu,2) + \\varepsilon \\mu(B_\\varepsilon(v)) \\mu(B_\\varepsilon(w))\\\\\n&\\geq \\PFP(\\mu,2) +\\varepsilon \\delta_\\varepsilon^2 > \\PFP(\\mu,2),\n\\end{align*}\n which is a contradiction. Thus, we have verified that $|\\langle x,y\\rangle|\\in \\{0,1\\}$, for all $x,y\\in \\supp(\\mu)$. Distinct elements in $\\supp(\\mu)$ are then either orthogonal to each other or antipodes. According to Corollary \\ref{theorem:tight p frame is necessary}, $\\supp(\\mu)$ is complete in $\\R^d$. Thus, there must be an orthonormal basis $\\{x_i\\}_{i=1}^d$ such that\n \\begin{equation*}\n  \\{x_1,\\ldots,x_d\\} \\subset \\supp(\\mu) \\subset \\{\\pm x_1,\\ldots,\\pm x_d\\}.\n \\end{equation*} \nConsequently, there is a density $f:S^{d-1}\\rightarrow\\R$ that vanishes on $S^{d-1}\\setminus \\supp(\\mu)$ such that $\\mu(x)=f(x)\\nu_{\\pm x_1,\\ldots,\\pm x_d}(x)$.  \n \nTo verify that $f$ satisfies (ii), let us define $\\tilde{f}:S^{d-1}\\rightarrow \\R$ by \n\\begin{equation*}\n\\tilde{f}(x)=\\begin{cases} f(x)+f(-x),& x\\in\\{x_1,\\ldots,x_d\\}\\\\\n0,& \\text{ otherwise. } \n\\end{cases}\n\\end{equation*}\nThis implies that $\\tilde{\\mu}(x)=\\tilde{f}(x)\\nu_{x_1,\\ldots,x_d}(x)$ is also a minimizer of $\\PFP(\\cdot,2)$. But the minimizers of the  probabilistic frame potential for $p=2$ have been investigated in~\\cite[Section 3]{Ehler:2010aa}. We can follow the arguments given there to obtain $\\tilde{f}(x_i)=\\frac{1}{d}$, for all $i=1,\\ldots,d$. \n\\end{proof}\n\n\n \n For even integers $p$, we can  give the minimum of $\\PFP(\\mu, p)$ and characterize its minimizers. The following theorem generalizes Theorem \\ref{theorem:p even integer discrete}. Moreover, note that the bounds are now sharp, i.e., for any even integer $p$, there is a probabilistic tight $p$-frame: \n \n \\begin{theorem}\\label{theorem:p even integer}\n Let $p$ be an even integer. For any probability distribution $\\mu$  on $S^{d-1}$,  \n  \\begin{equation*}\n \\PFP(\\mu, p)=\\int_{S^{d-1}}\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y) \\geq  \\frac{1\\cdot 3\\cdot 5\\cdots(p-1)}{d(d+2)\\cdots (d+p-2) },\n \\end{equation*}\nand equality holds if and only if $\\mu$ is a probabilistic tight $p$-frame. \n \\end{theorem}\n \n\n\n\n \\begin{proof}\n Let $\\alpha=\\frac{d}{2}-1$ and consider the Gegenbauer polynomials $\\{C_{n}^{\\alpha}\\}_{n\\geq 0}$ defined by \n \\begin{equation*}\n C_0^\\alpha(x)  = 1, \\qquad C_1^\\alpha(x) = 2 \\alpha x,\n \\end{equation*}\n \\begin{align*}\nC_{n}^\\alpha(x) &= \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)]\\\\\n&= C_n^{(\\alpha)}(z)=\\sum_{k=0}^{\\lfloor n\/2\\rfloor} (-1)^k\\frac{\\Gamma(n-k+\\alpha)}{\\Gamma(\\alpha)k!(n-2k)!}(2z)^{n-2k}.\n\\end{align*}\n$\\{C_{n}^{(\\alpha)}\\}_{n=1}^s$ is an orthogonal basis for the collection of polynomials of degree less or equal to $s$ on the interval $[-1,1]$ with respect to the weight\n\\begin{equation*}\nw(z) = \\left(1-z^2\\right)^{\\alpha-\\frac{1}{2}},\n\\end{equation*} \ni.e., for $m\\neq n$,\n \\begin{equation*}\n \\int_{-1}^1 C_n^{(\\alpha)}(x)C_m^{(\\alpha)}(x)w(x)\\,dx = 0.\n \\end{equation*}\n They are normalized by\n \\begin{equation*}\n \\int_{-1}^1 \\left[C_n^{(\\alpha)}(x)\\right]^2(1-x^2)^{\\alpha-\\frac{1}{2}}\\,dx = \\frac{\\pi 2^{1-2\\alpha}\\Gamma(n+2\\alpha)}{n!(n+\\alpha)[\\Gamma(\\alpha)]^2}.\n \\end{equation*}\nThe polynomials $t^p$, $p$ an even integer, can be represented by means of\n\\begin{equation*}\nt^p=\\sum_{k=0}^p \\lambda_k C^{\\alpha}_k(t).\n\\end{equation*}\nIt is known (see, e.g.,~\\cite{Bachoc:2005aa,Delsarte:1977aa}) that $\\lambda_i> 0$, $i=0,\\ldots,p$, and $\\lambda_0$ is given by\n \\begin{equation*}\n\\lambda_0= \\frac{1}{c}\\int_{-1}^1 t^p w(t) dt,\n \\end{equation*}\n where \n \\begin{equation*}\n c = \\frac{\\pi 2^{d+3}\\Gamma(d-2) }{(\\frac{d}{2}-1)\\Gamma(\\frac{d}{2}-1)^2}.\n \\end{equation*}\n Moreover,  $C^\\alpha_k$ induces a positive kernel, i.e., for $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ and $\\{u_i\\}_{i=1}^N\\subset \\R$,\n \\begin{equation*\n \\sum_{i,j=1}^{N} u_iC^{\\alpha}_k (\\langle x_i,x_j\\rangle )u_j \\geq  0, \\quad \\forall k=0,1,2,...\n \\end{equation*}\n see~\\cite{Bachoc:2005aa,Delsarte:1977aa}. Note that the probability measures with finite support are weak star dense in $\\mathcal{M}(S^{d-1},\\mathcal{B})$. Since $C^{\\alpha}_k$ is continuous, we obtain, for all $\\mu\\in \\mathcal{M}(S^{d-1},\\mathcal{B})$, \n \\begin{equation*} \n \\int_{S^{d-1}} \\int_{S^{d-1}} C^{\\alpha}_k (\\langle x,y\\rangle )d\\mu(x) d\\mu(y) \\geq  0, \\quad \\forall k=0,1,2,...\n \\end{equation*}\nWe can then estimate\n\\begin{align*}\n\\int_{S^{d-1}} \\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y) & = \\int_{S^{d-1}} \\int_{S^{d-1}}\\sum_{k=0}^p \\lambda_k C^{\\alpha}_k (\\langle x,y\\rangle )d\\mu(x) d\\mu(y)\\\\\n& = \\sum_{k=0}^p \\lambda_k \\int_{S^{d-1}} \\int_{S^{d-1}}C^{\\alpha}_k (\\langle x,y\\rangle ) d\\mu(x) d\\mu(y) \\geq \\lambda_0.\n\\end{align*}\nFrom the results in \\cite{Seidel:2001aa}, one can deduce that \n \\begin{equation*}\n\\lambda_0=  \\frac{1\\cdot 3\\cdot 5\\cdots(2t-1)}{d(d+2)\\cdots (d+2t-2) },\n \\end{equation*}\nwhich provides the desired estimate.\n\nWe still have to address the ``if and only if'' part. Equality holds if and only if $\\mu$ satisfies \n \\begin{equation*}\n  \\int_{S^{d-1}}\\int_{S^{d-1}} C^{\\alpha}_k (\\langle x,y\\rangle ) d\\mu(x) d\\mu(y) = 0, \\quad \\forall k=1,\\ldots, p. \n \\end{equation*}\nWe shall follow the approach outlined in \\cite{Venkov:2001aa} in which the analog  of Theorem~\\ref{theorem:p even integer discrete} was addressed  for finite symmetric collections of points. In this case, the finite symmetric sets of points lead to finite sums rather than integrals as above. The key ideas that we need in order to use the approach presented in \\cite{Venkov:2001aa} are: First, $\\tilde{\\mu}(E):=\\frac{1}{2}(\\mu(E)+\\mu(-E))$, for $E\\in\\mathcal{B}$, satisfies $\\PFP(\\tilde{\\mu},p) = \\PFP(\\mu,p)$. Thus, we can assume that $\\mu$ is symmetric. Secondly and more critically, the map \n\\begin{equation*}\ny\\mapsto \\int_{S^{d-1}} |\\langle x,y\\rangle |^p d\\mu(x)\n\\end{equation*}\nis a polynomial in $y$. In fact, the integral resolves in the polynomial's coefficients. These two observations enable us to follow the lines in \\cite{Venkov:2001aa}, and we can conclude the proof.\n\\end{proof}\n \n \\begin{remark}\nOne may speculate that Theorem \\ref{theorem:p even integer} could be extended to $p\\geq 2$ that are not even integers. This is not true in general. For $d=2$ and $p=3$, for instance, the equiangular FUNTF with $3$ elements induces a smaller potential than the uniform distribution. The uniform distribution is a probabilistic tight $3$-frame, but the equiangular FUNTF is not.\n\\end{remark}\n\n\n\n\n\\section*{Acknowledgements}\nThe authors would like to thank C.~Bachoc, W.~Czaja, C.~Wickman, and W.~S.~Yu for discussions leading to some of the results presented here. M.~Ehler was supported by the Intramural Research Program of the National Institute of Child Health and Human Development and by NIH\/DFG Research Career Transition Awards Program (EH 405\/1-1\/575910).  K.~A.~Okoudjou was partially supported by ONR grant N000140910324, by RASA from the Graduate School of UMCP, and by the Alexander von Humboldt foundation. \n\n\n\n\n\\bibliographystyle{plain}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\thispagestyle{empty}\n\nOne equivalent characterization of the amenability of an infinite group $G$, called the \\textit{F{\\o}lner condition}, is that the isoperimetric constant (also known as Cheeger constant) of its Cayley graph should be $0$.\nThat constant is defined as the infimum of $\\frac{|\\partial F|}{|F|}$ over all finite sets $F\\subset G$ with $|F|\\leq\\frac{1}{2}|G|$.\nAs the quotient cannot reach $0$, amenability of infinite groups is therefore characterized by the existence of a sequence of sets $F_n$ such that $\\frac{|\\partial  F_n|}{|F_n|}$ converges towards $0$, also known as a \\textit{F{\\o}lner sequence}.\nOne natural direction for studying the possible F{\\o}lner sequences on a given group is to ask how small the sets can be.\nWe consider the F{\\o}lner function.\nIt has classically been defined using the inner boundary:\n\\begin{equation}\\label{defdin}\n\t\\partial_{in}F=\\left\\{g\\in F:\\exists s\\in S\\bigcup S^{-1}:gs\\notin F\\right\\}.\n\\end{equation}\n\\begin{defi}\\label{foldef}\n\tThe \\textit{F{\\o}lner function} $\\Fol$ (or $\\Fol_S$; or $\\Fol_{G,S}$) of a group $G$ with a given finite generating set $S$ is defined on $\\N$ by\n\t$$\\Fol(n)=\\min\\left(|F|:F\\subset G,\\frac{|\\partial_{in}F|}{|F|}\\leq\\frac{1}{n}\\right).$$\n\\end{defi}\n\nRemark that $\\Fol(1)=1$ and that the values of the function are finite if and only if $G$ is amenable.\nIts values clearly depend on the choice of a generating set, but the functions arising from different generating sets (and more generally, functions arising from quasi-isometric spaces) are asymptotically equivalent.\nTwo functions are asymptotically equivalent if there are constants $A$ and $B$ such that $f(x\/A)\/B<g(X)<f(xA)B$.\nVarious articles have classified F{\\o}lner functions up to asymptotic equivalence, see for example~\\cite{BrieusselZheng,Erschler2003,Erschler2017,Gromov2009,Osserman1978,Pittet1995,Pittet2003,saloffcostezheng}.\nIn this paper, we will determine its exact values for one of the most basic examples of groups with exponential growth (we state our results in Section~\\ref{statesect}).\n\nThe classical isoperimetric theorem states that among domains of given volume in $\\R^n$, the minimal surface area is obtained on a ball (see survey by Osserman~\\cite[Section~2]{Osserman1978}).\nThe fact that if a minimum exists, it is realized only on the ball is obtained (in $\\R^2$) by Steiner in the XIX\\textsuperscript{th} century, using what is now called Steiner symmetrization (see Hehl~\\cite{hehl13}, Hopf~\\cite{hopf40}, Froehlich~\\cite{froehlich09}).\nThe existence of a minimum is obtained, in $\\R^3$, by Schwarz~\\cite{Schwarz1884}.\nAs $\\Z^n$ is quasi-isometric to $\\R^n$, this is also a first isoperimetric result for discrete groups.\nVaropoulos~\\cite{Varopoulis1985} shows more generally an isoperimetric inequality for direct products.\nPansu~\\cite{pansu83} (see also~\\cite{pansu82}) obtains one for the Heisenberg group $H_3$.\nOne central result is the Coulhon and Saloff-Coste inequality~\\cite{Coulhon1993}:\n\n\\begin{thm}[Coulhon and Saloff-Coste inequality]\\label{csc}\nConsider a group $G$ generated by a finite set $S$ and let $\\phi(\\lambda)=\\min(n:V(n)>\\lambda)$.\nThen for all finite sets $F$:\n\n$$\\frac{|\\partial_{in}F|}{|F|}\\geq\\frac{1}{8|S|\\phi(2|F|)}.$$\n\\end{thm}\nThe multiplicative constants can improved (see G\u00e1bor Pete~\\cite[Theorem~5.11]{pete}, Bruno Luiz Santos Correia~\\cite{csc2020}):\n\\begin{equation}\\label{csceq}\n\t\\frac{|\\partial_{in}F|}{|F|}\\geq\\frac{1}{2\\phi(2|F|)}.\n\\end{equation}\nThe result of Santos Correia is also announced for finite groups for $|F|\\leq\\frac{1}{2}|G|$.\nSantos Correia and Troyanov~\\cite{csc2021} show:\n\\begin{equation}\\label{csceqlam}\n\t\\frac{|\\partial_{in}F|}{|F|}\\geq\\left(1-\\frac{1}{\\lambda}\\right)\\frac{1}{\\phi(\\lambda|F|)}\n\\end{equation}\nfor $1<\\lambda\\leq\\frac{|G|}{|S|}$ (in particular, for arbitrarily large $\\lambda$ if $G$ is infinite).\nThe result is replicated in~\\cite{csccollab} without an upper bound on $\\lambda$.\n\nThe Coulhon and Saloff-Coste inequality (Theorem~\\ref{csc}) implies in particular that for a group with exponential growth, the F{\\o}lner function must also grow at least exponentially.\nIt can be obtained then that it is exactly exponential if one can describe F{\\o}lner sets with exponential growth.\nOne simple example is the lamplighter group $\\Z\\wr\\Z\/2\\Z$ with the standard generating set (\\ref{defst}).\nSimilarly, it is known that the F{\\o}lner functions of groups with polynomial growth are polynomial (see for example~\\cite[Section~I.4.C]{woess2000random}).\nAnother inequality on group isoperimetry is given by \u017buk~\\cite{Zuk2000}.\nVershik~\\cite{vershik1973countable} asks if F{\\o}lner function can be super-exponential, initiating the study of F{\\o}lner functions.\nHe suggests studying the wreath product $\\Z\\wr\\Z$ as a possible example.\nPittet~\\cite{Pittet1995} shows that the F{\\o}lner functions of polycyclic groups are at most exponential (and are therefore exponential for polycyclic groups with exponential growth).\nThis is true more generally for solvable groups of finite Pr\u00fcfer rank, see~\\cite{Pittet2003} and~\\cite{Kropholler2020}.\nThe first example of a group with super-exponential F{\\o}lner function is obtained by Pittet and Saloff-Coste~\\cite{PittetSaloffCoste} for $\\Z^d\\wr\\Z\/2\\Z$ with $d\\geq3$.\nLater the F{\\o}lner functions of wreath products with certain regularity conditions are described by Erschler~\\cite{Erschler2003} up to asymptotic equivalence.\nSpecifically, say that a function $f$ verifies property $(*)$ if for all $C>0$ there is $k>0$ such that $f(kn)>Cf(n)$.\nThe result of \\cite{Erschler2003} than states that if the F{\\o}lner function of a group $A$ verifies property $(*)$ (for some fixed generating set), then for any non-trivial group $B$, the F{\\o}lner function of $A\\wr B$ is $\\Fol_{A\\wr B}(n)=\\Fol_B(n)^{\\Fol_A(n)}$.\n\nOther examples with know F{\\o}lner functions have been presented by Gromov~\\cite[Section~8.2,Remark~(b)]{Gromov2009} for all functions with sufficiently fast growing derivatives.\nSaloff-Coste and Zheng~\\cite{saloffcostezheng} provide upper and lower bounds for it on, among others, \"bubble\" groups and cyclic Neumann-Segal groups, and those two bounds are asymptotically equivalent under certain conditions.\nRecently, Brieussel and Zheng~\\cite{BrieusselZheng} show that for any function $g$ that can be written as the inverse function of $x\/f(x)$ for some non-decreasing $f$ with $f(1)=1$ and $x\/f(x)$ also non-decreasing, there is a group the F{\\o}lner function of which is asymptotically equivalent to $\\exp(g(n))$.\nErschler and Zheng~\\cite{Erschler2017} obtain examples for a class of super-exponential functions under $\\exp(n^2)$ with weaker regularity conditions.\nSpecifically, for any $d$ and any non-decreasing $\\tau$ such that $\\tau(n)\\leq n^d$, there is a group $G$ and a constant $C$ such that\n\\begin{equation}\\label{annatyani}\n\tCn\\exp(n+\\tau(n))\\geq\\Fol_G(n)\\geq\\exp(\\frac{1}{C}(n+\\tau(n\/C))).\n\\end{equation}\nThe left-hand side of this inequality is always asymptotically equivalent to $\\exp(n+\\tau(n))$, and it suffices therefore that the right-hand side be asymptotically equivalent to that function to have a description of the F{\\o}lner function of $G$.\nNotice in particular that if $\\tau$ verifies condition $(*)$, this is verified.\nRemark that the conditions we mentioned only consider functions at least as large as $\\exp(n)$; it is an open question whether a F{\\o}lner function can have intermediate growth (see Grigorchuk~\\cite[Conjecture~5(ii)]{grigsurvey}).\nA negative answer would imply the Growth Gap Conjecture~\\cite[Conjecture~2]{grigsurvey}, which conjectures that the volume growth function must be either polynomial or at least as fast as $\\exp(\\sqrt{n})$.\nThose conjectures also have weak versions, which are equivalent to each other (see discussion after Conjecture~6 in~\\cite{grigsurvey}).\n\n\\section{Statement of results}\\label{statesect}\n\nIn this paper, we obtain the exact values of the F{\\o}lner function for the standard generating set $S=\\{t,\\delta\\}$ (see~(\\ref{defst})) on the lamplighter group $\\Z\\wr\\Z_2$ (see Definition~\\ref{deflamp}) where by $\\Z_2$ we denote $\\Z\/2\\Z$.\n\n\\begin{thm}\\label{thmmain}\nFor $n\\geq2$, the F{\\o}lner function of the lamplighter group $\\Z\\wr\\Z_2$ is, for the standard generating set: \n$$\\Fol(n)=2n2^{2(n-1)}.$$\n\\end{thm}\n\nWe also describe the sets that give rise to this function.\nSpecifically, we obtain that the standard F{\\o}lner sets $F_n=\\{(k,f):k\\in[\\![1,n]\\!],\\supp(f)\\subset[\\![1,n]\\!]\\}$ are optimal (see Definition~\\ref{optimal}) for the outer and edge boundary (see Section~\\ref{prel1}).\nWe then show that by Lemma~\\ref{equiv}, $F_n\\bigcup\\partial_{out}F_n$ is optimal for the inner boundary, from which Theorem~\\ref{thmmain} follows:\n\n\\begin{thm}\\label{thmlamp}\nConsider the lamplighter group $\\Z\\wr\\Z_2$ with the standard generating set.\n\\begin{enumerate}\n\t\\item For any $n\\geq2$ and any $F\\subset\\Z\\wr\\Z_2$ such that $|F|\\leq|F_n|$, we have\n\t$$\\frac{|\\partial_{edge}F|}{|F|}\\geq\\frac{|\\partial_{out}F|}{|F|}\\geq\\frac{|\\partial_{out}F_n|}{|F_n|}=\\frac{|\\partial_{edge}F_n|}{|F_n|},$$\n\tand if $|F|<|F_n|$, the inequality $\\frac{|\\partial_{out}F|}{|F|}>\\frac{|\\partial_{out}F_n|}{|F_n|}$ is strict,\n\t\\item From point $(1)$ it follows that for any $n\\geq2$ and any $F\\subset\\Z\\wr\\Z_2$ such that $|F|\\leq|F_n\\bigcup\\partial_{out}F_n|$, we have\n\t$$\\frac{|\\partial_{in}F|}{|F|}\\geq\\frac{|\\partial_{in}(F_n\\bigcup\\partial_{out}F_n)|}{|F_n\\bigcup\\partial_{out}F_n|},$$\n\tand if $|F|<|F_n\\bigcup\\partial_{out}F_n|$, the inequality is strict.\n\\end{enumerate}\nFurthermore, the sets that give equality are unique up to translation.\n\\end{thm}\n\nWe can substitute those values in the Coulhon and Saloff-Coste inequality in order to study the multiplicative constant.\nAs in~\\cite{csccollab}, we define\n\\begin{defi}\\label{constq}\nFor a group $G$ and a generating set $S$, denote\n$$C_{G,S}=\\sup\\left\\{c\\geq0:\\exists\\alpha\\geq0\\mbox{ such that }\\forall F\\subset G,\\frac{|\\partial_{in}F|}{|F|}\\geq c\\frac{1}{\\phi((1+\\alpha)|F|)}\\right\\},$$\nwhere $F$ is assumed to be finite and non-empty.\n\\end{defi}\nThe original inequality obtains that for all $G,S$, $C_{G,S}\\geq\\frac{1}{8|S|}$.\nThe results of \\cite[Theorem~5.11]{pete} and \\cite{csc2020} that we cited as Equation~\\ref{csceq} give a lower bound of $\\frac{1}{2}$.\nEquation~\\ref{csceqlam} from~\\cite{csc2021} further implies that $C_{G,S}\\geq1$ for all $G,S$.\n\nIn \\cite{csccollab}, it is shown that for groups of exponential growth:\n$$C_{G,S}=\\frac{\\liminf\\frac{\\ln\\Fol(n)}{n}}{\\lim\\frac{\\ln V(n)}{n}}.$$\n\\begin{prop}\\label{const}\nThe lamplighter group verifies\n\n$$C_{\\Z\\wr\\Z_2,S}=\\frac{\\lim\\frac{\\ln\\Fol(n)}{n}}{\\lim\\frac{\\ln V(n)}{n}}=\\frac{\\ln4}{\\ln(\\frac{1}{2}(1+\\sqrt{5}))}\\approx2,88$$\nfor the standard generating set. \n\\end{prop}\n\n\\begin{remark}\\label{const2}\nFor the switch-walk-switch generating set $S'=\\{t,\\delta,t\\delta,\\delta t,\\delta t\\delta\\}$, we have\n\n$$C_{\\Z\\wr\\Z_2,S'}=\\frac{\\liminf\\frac{\\ln\\Fol_{sws}(n)}{n}}{\\lim\\frac{\\ln V_{sws}(n)}{n}}\\leq2.$$\n\\end{remark}\n\nAnother direction that can be considered once one has exact evaluations of a F{\\o}lner function is studying the power series $\\sum_n\\Fol(n)x^n$.\nThe equivalent series have been studied for volume growth (see Grigorchuk-de la Harpe~\\cite[Section~(4)]{Grigorchuk1997}).\nOne central question that a lot of authors have considered is the rationality of those series as a function.\nFor the example shown here, the power series of the F{\\o}lner function is a rational function:\n$$\\sum_{n\\in\\N}\\Fol(n)x^n=\\frac{2x}{(4x-1)^2}.$$\n\nWe also obtain results for the Baumslag-Solitar group $BS(1,2)$ (see Definition~\\ref{defbs}), however only in respect to the edge boundary.\nTaking the notation from the definition, its standard sets are defined the same way as in the lamplighter group.\n\\begin{thm}\\label{bsthm}\nConsider the Baumslag-Solitar group $BS(1,2)$ with the standard generating set.\nThen for any $n\\geq2$ and any $F\\subset BS(1,2)$ such that $|F|\\leq|F_n|$, we have $\\frac{|\\partial_{edge}F|}{|F|}\\geq\\frac{|\\partial_{edge}F_n|}{|F_n|}$ (where $F_n$ are the standard F{\\o}lner sets), and if $|F|<|F_n|$, the inequality is strict.\n\\end{thm}\nThis result is not always true for $B(1,p)$ for larger $p$, and we provide a counter example in Example~\\ref{exbsp}.\nHowever this counter example uses that $p$ is significant when compared to the length of the interval defining the standard set, and it is possible that for $B(1,p)$ as well, standard sets are optimal above a certain size.\n\nWe present more detailed definitions in the next section.\nIn Section~\\ref{prelim2}, we present associated graphs, which are the main tool of the proof, and prove some general results.\nIn particular, we show Lemma~\\ref{equiv}, which will be used to obtain that part $(2)$ of Theorem~\\ref{thmlamp} follows from part $(1)$.\nIn Section~\\ref{mainsection}, we prove Theorem~\\ref{thmlamp}.\nIn Section~\\ref{csc-const-sect}, we prove Proposition~\\ref{const} and Remark~\\ref{const2}.\nFinally, in Section~\\ref{bssect}, we prove Theorem~\\ref{bsthm} and Example~\\ref{exbsp}.\n\n\\section{Preliminaries}\\label{prel1}\n\nThe concept of amenability finds its origins in a 1924 result by Banach and Tarski~\\cite{Banach-Tarski-original}, where they decompose a solid ball in $\\R^3$ into five pieces, and reassemble them into two balls using rotations.\nThat is now called the Banach-Tarski paradox.\nThe proof makes use of the fact that the group of rotations of $\\R^3$ admits a free subgroup.\nVon Neumann~\\cite{Neumann1929} considers it as a group property and introduces the concept of amenable groups.\nNowadays, there are multiple different characterizations of amenability; see books by Greenleaf~\\cite{greenleaf} and Wagon~\\cite{Banach-Tarski}, or an article by Ceccherini-Silberstein-Grigorchuk-la~Harpe~\\cite{MR1721355}, or a recent survey by Bartholdi~\\cite{bartholdi}.\n\n\\begin{defi}[F{\\o}lner criterion]\\label{folamdef}\n\tA group $G$ is amenable if and only if for every finite set $S\\subset G$ and every $\\varepsilon>0$ there exists a set $F$ such that\n\t\n\t$$|F\\Delta F.S|\\leq\\varepsilon|F|.$$\n\\end{defi}\n\nIf $G$ is finitely generated, it suffices to consider a single generating set $S$ instead of all finite sets.\nWe can also apply Definition~\\ref{folamdef} for $S\\bigcup S^{-1}\\bigcup\\{\\Id\\}$.\nThen $|F\\Delta(S\\bigcup S^{-1}\\bigcup\\{\\Id\\}).F|$ is the set of vertices in the Cayley graph of $G$ that are at a distance exactly $1$ from $F$.\nWe denote that the outer boundary $\\partial_{out}F$.\nThen the condition can be written as $\\frac{|\\partial_{out}F_n|}{|F_n|}\\leq\\varepsilon$, or in other words that the  infimum of those quotients should be $0$.\nRecall the definition (\\ref{defdin}) of the inner boundary.\nFinally, we consider $\\partial_{edge}F$ to be the set of edges between $F$ and its complement.\nRemark that while those values can differ, whether the infimum of $\\frac{|\\partial F|}{|F|}$ is $0$ or not does not depend on which boundary we consider.\n\nFor groups of subexponential growth, for every $\\varepsilon$, there is some $n$ such that the ball around the identity of radius $n$ is a corresponding F{\\o}lner set.\nNote that to obtain a F{\\o}lner sequence from this, one needs to consider a subsequence of the sequence of balls of radius $n$.\nIt is an open question whether in every group of subexponential growth, all balls form a F{\\o}lner sequence.\nFor groups of exponential growth, it is generally not sufficient to consider balls, and it is an open question whether there exists any group of exponential growth where some subsequence of balls forms a F{\\o}lner sequence (see for example Tessera~\\cite[Question~15]{Tessera2007}).\n\nFor two groups $A$ and $B$ and a function $f\\in B^A$, denote\n$$\\supp(f)=\\{a\\in A:f(a)\\neq\\Id_B\\}.$$\nLet $B^{(A)}$ be the set of functions from $A$ onto $B$ with finite support.\n\\begin{defi}\\label{deflamp}\n\tThe (restricted) wreath product $A\\wr B$ is the semidirect product $A\\ltimes B^{(A)}$ where $A$ acts on $B^{(A)}$ by translation.\n\\end{defi}\nWe can write the elements as $(a,f)$ with $a\\in A$ and $f\\in B^{(A)}$.\nThe group law is then $(a,f)(a',f')=(aa',x\\mapsto f(x)f'(a^{-1}x))$.\n\nGiven generating sets $S$ and $S'$ on $A$ and $B$ respectively, we can define a standard generating set on $A\\wr B$.\nIt consists of the elements of the form $(s,\\mathbb{Id_B})$ for $s\\in S$ (where $\\mathbb{Id_B}=\\Id_B$ for all $x\\in A$), as well as $(\\Id_A,\\delta_{\\Id_A}^{s'})$ for $s'\\in S'$ where $\\delta_{\\Id_A}^{s'}(\\Id_A)=s'$ and $\\delta_{\\Id_A}^{s'}(x)=\\Id_B$ for all other $x$.\nOne can verify that $(a,f)(s,\\mathbb{Id_B})=(as,f)$, and $(a,f)(\\Id_A,\\delta_{\\Id_A}^{s'})=(a,f+\\delta_a^{s'})$, or in other words the value of $f$ at the point $a$ is changed by $s'$.\n\nSimilarly, given F{\\o}lner sets $F_A$ and $F_B$ on $A$ and $B$ respectively, one obtains standard F{\\o}lner sets on $A\\wr B$: \n$$F=\\{(a,f):a\\in F_A,\\supp(f)\\subset F_A,\\forall x:f(x)\\in F_B\\}.$$\nTheir outer boundary is\n\\begin{equation*}\n\\begin{split}\n\\partial_{out}F&=\\{(a,f):a\\in\\partial_{out}F_A,\\supp(f)\\subset F_A,\\forall x:f(x)\\in F_B\\}\\\\\n&\\cup\\{(a,f):a\\in F_A,\\supp(f)\\subset F_A,f(a)\\in\\partial_{out}F_B,\\forall x\\neq a:f(x)\\in F_B\\}.\n\\end{split}\n\\end{equation*}\nAs $|F|=|F_A||F_B|^{|F_A|}$ and $|\\partial_{out}F|=|\\partial_{out}F_A||F_B|^{|F_A|}+|F_A||F_B|^{|F_A|-1}|\\partial_{out}F_B|$, we have\n$$\\frac{|\\partial_{out}F|}{|F|}=\\frac{|\\partial_{out}F_A|}{|F_A|}+\\frac{|\\partial_{out}F_B|}{|F_B|}.$$\n\nWe will focus on the lamplighter group $\\Z\\wr\\Z_2$.\nAs both of those groups have standard generating sets, this gives us a standard generating set on the lamplighter group:\n\n\\begin{equation}\\label{defst}\nS=\\{t,\\delta\\}\\mbox{ where }t=(1,\\mathbbold{0})\\mbox{ and }\\delta=(0,\\delta^1_0).\n\\end{equation}\n\nThe Baumslag-Solitar groups are defined as follows:\n\\begin{defi}\\label{defbs}\n\tThe Baumslag-Solitar group $BS(m,n)$ is the two-generator group given by the presentation $\\langle a,b:a^{-1}b^ma=b^n\\rangle$.\n\\end{defi}\nThe standard generating set is $\\{a,b\\}$.\n\nWe will focus on the groups $BS(1,p)$.\nThat group is isomorphic to the group generated by $x\\mapsto px$ and $x\\mapsto x+1$ (by mapping $a^{-1}$ and $b$ to them respectively).\nBy abuse of notation, we will also denote the images of $a$ and $b$ with the same letters.\nIn that group, any element can be written as $x\\mapsto p^nx+f$ with $n\\in\\Z$ and $f\\in\\Z[\\frac{1}{p}]$.\nWe then have $(x\\mapsto p^nx+f)a=x\\mapsto p^{n-1}x+f$ and $(x\\mapsto p^nx+f)b=x\\mapsto p^nx+(f+p^n)$.\n\nRemark that the subgroup $N=\\{x\\mapsto x+f:f\\in\\Z[\\frac{1}{p}]\\}$ is normal.\nIndeed, we have $(x\\mapsto p^{-n}(x-f))\\circ(x\\mapsto p^nx+f)=\\Id$ and\n$$(x\\mapsto p^{-n}(x-f'))\\circ(x\\mapsto x+f)\\circ(x\\mapsto p^nx+f')=x\\mapsto x+p^nf.$$\nThus $BS(1,p)$ is isomorphic to the semidirect product $\\Z\\ltimes\\Z[\\frac{1}{p}]$ defined by the action $n.f=p^nf$.\nWe therefore write the element $x\\mapsto p^nx+f$ as $(n,f)$.\nThe standard F{\\o}lner sets are then expressed in the same way as for wreath products.\nIn other words:\n$$F_n=\\{(k,f):k\\in[\\![0,n-1]\\!],f\\in\\Z,0\\leq f<p^n\\}.$$\n\n\\section{Main concepts of the proof}\\label{prelim2}\n\n\\begin{defi}\\label{optimal}\n\tWe will call a set $F$ in a group $G$ \\textit{optimal} with respect to the inner (respectively outer, edge) boundary if for any $F'$ with $|F'|\\leq|F|$, it is true that $\\frac{|\\partial_{in}F'|}{|F'|}\\geq\\frac{|\\partial_{in}F|}{|F|}$ (respectively $\\frac{|\\partial_{out}F'|}{|F'|}\\geq\\frac{|\\partial_{out}F|}{|F|}$, $\\frac{|\\partial_{edge}F'|}{|F'|}\\geq\\frac{|\\partial_{edge}F|}{|F|}$), and if $|F'|<|F|$, the inequality is strict.\n\\end{defi}\n\n\\begin{lemma}\\label{equiv}\nIf $F$ is optimal for the outer boundary, up to replacing $F$ with another optimal set of the same size, $F\\bigcup\\partial_{out}F$ is optimal for the inner boundary.\nFurthermore, this describes all optimal sets of that size.\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the large inequality.\nThe case for the strict inequality is equivalent.\n\nLet $F$ be optimal for the outer boundary and consider $F'$ such that $|F'|\\leq|F\\bigcup\\partial_{out}F|$ and the quotient of the inner boundary is smaller.\nWithout loss of generality, we can assume that $F'$ is optimal for the inner boundary.\n\nLet $F''=F'\\setminus\\partial_{in}F'$.\nObserve that $\\partial_{out}F''\\subset\\partial_{in}F'$.\nWe first claim that $F'$ being optimal implies that $\\partial_{out}F''=\\partial_{in}F'$.\nIndeed, $|F''\\bigcup\\partial_{out}F''|\\leq|F'|$, and\n$$\\frac{|\\partial_{in}(F''\\bigcup\\partial_{out}F'')|}{|F''\\bigcup\\partial_{out}F''|}\\leq\\frac{|\\partial_{out}F''|}{|F''\\bigcup\\partial_{out}F''|}=\\frac{\\frac{|\\partial_{out}F''|}{|F''|}}{1+\\frac{|\\partial_{out}F''|}{|F''|}},$$\nwhile\n$$\\frac{|\\partial_{in}F'|}{|F'|}=\\frac{\\frac{|\\partial_{in}F'|}{|F''|}}{1+\\frac{|\\partial_{in}F'|}{|F''|}}.$$\nAs $\\partial_{out}F''\\subset\\partial_{in}F'$ and $\\frac{x}{x+1}$ is an increasing function in $\\R_+$, the first quantity is smaller then the second, and by $F'$ being optimal, we have an equality.\n\nWe now consider cases for the size of $F''$.\nIf $|F''|<|F|$, then we can apply the assumption that $F$ is optimal and we get\n$$\\frac{|\\partial_{out}F|}{|F|}\\leq\\frac{|\\partial_{out}F''|}{|F''|}=\\frac{|\\partial_{in}F'|}{|F'\\setminus\\partial_{in}F'|}=\\frac{\\frac{|\\partial_{in}F'|}{|F'|}}{1-\\frac{|\\partial_{in}F'|}{|F'|}}.$$\nHowever, applying the initial assumption by which we chose $F'$ gives us the inverse inequality, and strict.\n\nWe are left with the case where $|F''|\\geq|F|$.\nLet $k=|F''|-|F|$ and remove any $k$ points from $F''$ to obtain $F'''$.\nWe obtain a set that is the same size as $F$ and has an outer boundary no larger than that of $F$.\nIt is therefore another optimal set of the same size, and by the optimality of $F'$ for the inner boundary, $F'''\\bigcup\\partial_{out}F'''=F'$, which concludes the proof.\n\\end{proof}\n\nThe central idea of this paper is to work on an associated graph structure which we define for semidirect products.\n\\begin{defi}\\label{assgr}\nConsider a semidirect product $G=H\\ltimes N$.\nConsider a generating set $S$ of $G$.\nWe define the \\textit{associated graph} as the directed labeled graph $\\Gamma=\\Gamma_S$ with vertex set $V(\\Gamma)=N$ and edge set \n\t\n$$\\overrightarrow{E}(\\Gamma)=\\left\\{\\overrightarrow{(f_1,f_2)}:\\exists s\\in S,h_1,h_2\\in H\\mbox{ such that }h_1f_1s=h_2f_2\\right\\}.$$\nWith these notations, the edge $\\overrightarrow{(f_1,f_2)}$ is labeled $s$.\n\\end{defi}\nAs mentioned, the two examples we will consider here are the lamplighter group $\\Z\\wr\\Z_2$ and the Baumslag-Solitar group $BS(1,p)$.\nIn both examples we have $H=\\Z$.\nIn the case of the lamplighter group we have $N=\\Z_2^{(\\Z)}$, and for $BS(1,p)$, $N$ is the set of $p$-adic numbers.\n\nWe define an associating function $\\phi:G\\rightarrow\\mathrm{P}(\\Gamma)$ by \n$$\\phi(hf)=\\left\\{\\overrightarrow{(f,f')}:hfs=h'f'\\mbox{ for some }h'\\in H,s\\in S\\right\\}.$$\n\\begin{defi}\\label{asssubgr}\nConsider a semidirect product $G=H\\ltimes N$.\nLet $F$ be a finite subset of $G$.\nThe \\textit{associated subgraph} of $F$ is the subgraph of the associated graph $\\Gamma$ made of the edges\n$$\\bigcup_{x\\in F}\\phi(x)$$\nand all adjacent vertices.\n\\end{defi}\nWe will provide a bound for the boundary of a set based on a formula on the associated subgraph, and maximize the value of that formula over all subgraphs of $\\Gamma$ no larger (in terms of number of edges) than the associated subgraph.\n\n\\section{The lamplighter group}\\label{mainsection}\n\nIn this section we provide the proof of Theorem~\\ref{thmlamp}, the larger part being a proof of Theorem~\\ref{thmlamp}$(1)$.\nIn other words, we show that for the lamplighter group with the standard generating set, the standard sets are optimal with respect to the outer boundary, and uniquely so up to translation.\nWe first prove an inequality relying the isoperimetry of a subset with values on the associated graph.\n\n\\begin{lemma}\\label{main}\nLet $F$ be a finite set in $\\Z\\wr\\Z_2$ with $\\frac{|\\partial_{out}F|}{|F|}\\leq1$.\nLet $\\Gamma$ be the associated graph (see Definition~\\ref{assgr}).\nThen\n\n$$\\frac{|\\partial_{out}F|}{|F|}\\geq\\min\\left(\\frac{2|V(G)|}{|E(G)|}\\mbox{ for }G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|E(G)|\\leq|F|\\right).$$\n\\end{lemma}\nWe will obtain that by estimating the number of points in the boundary that are reachable from the set by multiplication by $t$ or $t^{-1}$.\n\n\\begin{proof}\nAs mentioned below the definition of associated graph, the vertex set of $\\Gamma$ is $\\Z_2^{(\\Z)}$.\nThe edge set is $\\left\\{\\overrightarrow{(f,g)}:\\exists x_0:f(x)=g(x)\\iff x\\neq x_0\\right\\}$, and all of them have the label $\\delta$.\nAs all edges have the same label, we will think of $\\Gamma$ as unlabeled.\t\n\nConsider a finite set $F$ of elements of $\\Z\\wr\\Z_2$.\nLet $\\widetilde{F}$ be the associated subgraph (see Definition~\\ref{asssubgr}).\nA \\textbf{leaf} we call a vertex which is included in exactly one edge of the subgraph and is at the tail of that edge.\nThe set of leaves we denote by $L(\\widetilde{F})$.\n\nWe would like to estimate, for each vertex $f$ in $\\widetilde{F}$, the number of points of the form $(n,f)$ in $\\partial_{out}F$.\nWe will show that if $f$ is a leaf, there is at least one such point, and if $f$ is not a leaf, there are at least two.\nThe first case follows directly from the definition of a leaf.\n\nAssume that $f$ is not a leaf.\nThen either there are at least two edges ending in $f$, or there is at least one edge starting at $f$.\nOnce again, the first case follows directly from the definition.\nIn the second case, there is an element $(k,f)\\in F$.\nLet $a=\\max\\{k:(k,f)\\in F\\}$ and $b=\\min\\{k:(k,f)\\in F\\}$.\nThen $(a+1,f)$ and $(b-1,f)$ are in the outer boundary of $F$.\n\nWe obtain:\n\n\\begin{equation}\\label{ineq}\n|\\partial_{out}F|\\geq2(|V(\\widetilde{F})|-|L(\\widetilde{F})|)+|L(\\widetilde{F})|=2|V(\\widetilde{F})|-|L(\\widetilde{F})|.\n\\end{equation}\n\nRemark that if there are no leaves, the desired result follows.\n\n\\begin{lemma}\\label{leaf}\nFix $n$.\nAssume that\n$$\\min\\left(\\frac{2|V(G)|-|L(G)|}{|E(G)|}:G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|L(G)|\\neq0\\mbox{ and }|E(G)|\\leq n\\right)\\leq1.$$\nThen\n\\begin{equation*}\n\\begin{split}\n&\\min\\left(\\frac{2|V(G)|-|L(G)|}{|E(G)|}:G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|L(G)|\\neq0\\mbox{ and }|E(G)|\\leq n\\right)\\\\\n&\\geq\\min\\left(\\frac{2|V(G)|}{|E(G)|}:G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|E(G)|\\leq n-1\\right).\n\\end{split}\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the lemma by induction on $n$.\nFor small enough $n$, the assumed inequality is impossible, which gives us the base of the induction.\n\nAssume that the statement is true for $n-1$ and consider $G$ with $n$ edges such that\n\n$$\\frac{2|V(G)|-|L(G)|}{|E(G)|}=\\min\\left(\\frac{2|V(H)|-|L(H)|}{|E(H)|}:|L(H)|\\neq0\\mbox{ and }|E(H)|\\leq n\\right)\\leq1.$$\n\nIf $G$ does not contain a leaf, the statement is trivial.\nWe now assume that $G$ contains a leaf.\nLet $G'$ be the subgraph obtained by removing that leaf and the edge leading to it.\nThen $|V(G')|=|V(G)|-1$ and $|E(G')|=n-1$.\n\nNotice also that while new leaves may have appeared, at most one was removed, and thus $|L(G')|\\geq|L(G)|-1$.\nTherefore, as $\\frac{2|V(G)|-|L(G)|}{|E(G)|}\\leq1$, we have:\n$$\\frac{2|V(G')|-|L(G')|}{|E(G')|}\\leq\\frac{2|V(G)|-|L(G)|-1}{|E(G)|-1}\\leq\\frac{2|V(G)|-|L(G)|}{|E(G)|}.$$\n\nIt is in particular less than $1$, and we obtain:\n$$\\frac{2|V(G')|-|L(G')|}{|E(G')|}\\geq\\min\\left(\\frac{2|V(H)|}{|E(H)|}:H\\mbox{ subgraph of }\\Gamma\\mbox{ with }|E(H)|\\leq n-1\\right),$$\neither from the induction hypothesis or directly, depending on whether $G'$ has a leaf or not.\n\\end{proof}\nThe result of Lemma~\\ref{main} then follows from equation (\\ref{ineq}) and applying Lemma~\\ref{leaf} for $n=|E(\\widetilde{F})|$.\n\\end{proof}\n\nHaving proven Lemma~\\ref{main}, we now need to show that the standard sets $F_n$ minimize $\\frac{2|V(G)|}{|E(G)|}$ over subgraphs of $\\Gamma$ with a fixed amount of edges.\nThe associated subgraph of $F_n$ (see Definition~\\ref{asssubgr}) is the subgraph of $\\Gamma$ with vertices $\\{f:\\supp(f)\\subset[\\![1,n]\\!]\\}$ and all edges of $\\Gamma$ between two of those vertices.\nIt has $2^n$ vertices and $2n2^n$ edges.\n\n\\begin{lemma}\nLet $G$ be a subgraph of $\\Gamma$ with at most $2n2^n$ edges.\nThen $\\frac{2|V(G)|}{|E(G)|}\\geq\\frac{1}{n}$, and if they are equal, $G$ is a hypercube (with double edges) with $2^n$ vertices.\n\\end{lemma}\n\n\\begin{proof}\nWe will first see that we can assume that any directed edge is present simultaneously with its inverse.\nIndeed, consider the set of directed edges $\\overrightarrow{(f,g)}$ in $G$ such that $\\overrightarrow{(g,f)}$ is not in $G$.\nIf there are at least two such edges, removing one and adding the inverse of the other changes neither $|V(G)|$ nor $|E(G)|$.\nIf there is exactly one such edge $\\overrightarrow{(f,g)}$, then $|E(G)|$ is odd, and so it is at most $2n2^n-1$.\nIn that case by adding the edge $\\overrightarrow{(g,f)}$ we obtain another subgraph with at most $2n2^n$ edges, and such that the quotient we consider is smaller.\n\nWe can therefore assume that any directed edge is present simultaneously with its inverse.\nWe replace every couple of edges $\\overrightarrow{(f,g)}$ and $\\overrightarrow{(g,f)}$ with one undirected edge $(f,g)$.\nWe will call the graphs we obtain $\\overline{\\Gamma}$ and $\\overline{G}$.\nWe need to prove that $\\frac{|V(\\overline{G})|}{|E(\\overline{G})|}\\geq\\frac{1}{n}$, and if they are equal, $\\overline{G}$ is a hypercube with $2^n$ vertices.\n\nRemark that $\\overline{\\Gamma}$ is the bi-infinite hypercube.\nAs $\\overline{G}$ is finite, it is contained in some finite hypercube.\nIn that case, the question of maximizing the number of edges on a fixed number of vertices has already been answered in literature.\nOne proof is presented in Harper's book~\\cite[Section~1.2.3]{Harper2004}.\nTaking notations from the book, consider a subset $C\\subset\\Z$ of cardinal $c$, and a vertex $f$ of $\\overline{\\Gamma}$ with $\\supp(f)\\bigcap C=\\emptyset$.\nThe set\n$$\\left\\{g\\in V(\\overline{\\Gamma}):f(x)=g(x)\\forall x\\notin C\\right\\}$$\nis called a \\textbf{$c$-subcube}.\nA vertex set $S$ of cardinal $k=\\sum_{i=1}^K2^{c_i}$ with $0\\leq c_i<c_j$ for $i<j$ is \\textbf{cubal} if it is a disjoint union of $c_i$-subcubes with the $c_i$-subcube being contained in the neighborhood of the $c_j$-subcube for all $i<j$.\nHere, neighborhood means the set of points at distance $1$ in graph distance.\nRemark that two cubal sets of the same cardinality are isomorphic.\n\nBy abuse of notation, for a set $S$ of vertices in $\\Gamma$, we will denote by $E(S)$ the set of all edges between two vertices of $S$.\nThen Theorem~$1.1$ from the cited section states\n\\begin{thm}[{\\cite[Section~1.2.3~-~Theorem~1.1]{Harper2004}}]\\label{thmharper}\n\t$S$ maximizes $|E(S)|$ over sets with cardinal $|S|$ if and only if $S$ is cubal.\n\\end{thm}\n\nAs $|E(S)|$ increases strictly with $|S|$, and equality can be achieved for $|E(S)|=n2^n$, it therefore suffices to consider the case where $\\overline{G}$ is cubal.\nLet $|V(\\overline{G})|=\\sum_{i=1}^K2^{c_i}$.\nIf $|V(\\overline{G})|>2^n$, the result follows immediately.\nAssume that $|V(\\overline{G})|\\leq2^n$, and thus $c_K\\leq n$, and $c_i\\leq n-K+i$.\nThen\n$$|E(\\overline{G})|=\\sum_{i=1}^Kc_i2^{c_i}+\\sum_{i<j}2^{c_i}=\\sum_{i=1}^{K}(c_i+K-i)2^{c_i}\\leq n\\sum_{i=1}^K2^{c_i},$$\nwhich concludes the proof of the inequality.\nFurthermore, if there is equality, then $c_K=n$, in which case $|V(\\overline{G})|=n2^n$ and $K=1$.\n\\end{proof}\n\nTheorem~\\ref{thmlamp}$(1)$ follows from this lemma and Lemma~\\ref{main}.\nBy Lemma~\\ref{equiv}, Theorem~\\ref{thmlamp}$(2)$ follows from Theorem~\\ref{thmlamp}$(1)$ (and the unicity of the optimal sets).\n\n\\section{Bounds for the Coulhon and Saloff-Coste inequality for the lamplighter group}\\label{csc-const-sect}\n\nWe will now prove Proposition~\\ref{const}.\nRecall its statement:\n\\begin{prop*}\nThe lamplighter group verifies\n\n$$C_{\\Z\\wr\\Z_2,S}=\\frac{\\lim\\frac{\\ln\\Fol(n)}{n}}{\\lim\\frac{\\ln V(n)}{n}}=\\frac{\\ln4}{\\ln(\\frac{1}{2}(1+\\sqrt{5}))}\\approx2,88$$\nfor the standard generating set.\n\n\\end{prop*}\n\n\\begin{proof}\nWe have obtained in Theorem~\\ref{thmmain} that $\\lim\\sqrt[n]{\\Fol(n)}=4$.\nIt is therefore sufficient to calculate the exponent of its volume growth.\n\nConsider first the standard generating set.\nConsider an element $g$ of length $n$ in the group and let its support be $[m,p]$.\nWe can write $g=t^mAt^i$ where $A$ is a non-reducible word on $t$ and $\\delta$ (without their inverses).\nIt is not hard to see that $A$ contains the letter $t$ exactly $p-m$ times.\nRemark that any representation of $g$ in the standard generators must contain $t$ or $t^{-1}$ at least $p-m$ times.\nAdditionally, it contains $\\delta$ at least as many times as $A$ does.\nTherefore $|A|\\leq|g|=n$.\n\nNotice also that $|m|\\leq n$ and $|i|\\leq2n$.\nTherefore, if we denote by $V'(n)$ the amount of non-reducible words on $t$ and $\\delta$ of length at most $n$, we obtain\n$$V'(n)8n^3\\geq V(n)\\geq V'(n).$$\n\nWe now seek to calculate $V'(n)$.\nNotice that the only condition on those words in the lamplighter group is to not have two consecutive $\\delta$-s.\nTherefore $V'(n)$ is same as the amount of subsets of $[1,n]$ without two consecutive elements.\nA simple induction shows that those are (up to translation) the Fibonacci numbers.\nIndeed, such a subset either has $n$ in it, in which case the rest of the subset is contained in $[1,n-2]$, or it does not have $n$ and the rest of the subset is contained in $[1,n-1]$.\nTherefore\n$$\\lim\\sqrt[n]{V(n)}=\\lim\\sqrt[n]{V'(n)}=\\frac{1+\\sqrt{5}}{2}.$$\n\\end{proof}\n\nIt is worth noting that the exact value of the volume growth power series $\\sum_nV(n)x^n$ for the standard generating set has been described by Parry~\\cite{Parry1992a}.\n\nWe then prove Remark~\\ref{const2}.\nIt states:\n\\begin{remark*}\nFor the switch-walk-switch generating set $S'=\\{t,\\delta,t\\delta,\\delta t,\\delta t\\delta\\}$, we have\n\n$$C_{\\Z\\wr\\Z_2,S'}=\\frac{\\liminf\\frac{\\ln\\Fol_{sws}(n)}{n}}{\\lim\\frac{\\ln V_{sws}(n)}{n}}\\leq2.$$\n\\end{remark*}\n\\begin{proof}\nThe standard F{\\o}lner sets verify that $\\frac{|\\partial'_{out}F_n|}{|F_n|}=\\frac{2}{n}$ where by $\\partial'_{out}$ we denote the outer boundary with regards to $S'$.\nTherefore $\\lim\\sqrt[n]{\\Fol_{sws}(n)}\\leq4$.\nSimilarly, we now need to calculate the exponent of its volume growth.\n\nThe switch-walk-switch volume is also controlled by the size of the set of possible configurations given a certain interval.\nHowever, in its case all configurations with support in that interval are found on an element in the ball.\nThus $8n^32^n\\geq V_{sws}(n)\\geq2^n$ and $\\lim\\sqrt[n]{V_{sws}(n)}=2$.\n\\end{proof}\n\n\\section{The Baumslag-Solitar group $BS(1,2)$}\\label{bssect}\n\nWe now prove Theorem~\\ref{bsthm}.\nRemark that for the edge boundary, the contributions by each element of the generating set are always disjoint.\nRecall that the vertex set of the associated graph $\\Gamma'$ is the dyadic numbers.\nFor the standard generating set, the edges are the couples $\\overrightarrow{(f,g)}$ such that $|f-g|$ is a power of $2$, and are labeled $b$.\nFor any finite subgraph of $\\Gamma'$, up to translation we can assume that its vertices are all integers, in which case edges will be defined by positive powers of $2$.\n\nWe introduce the following notations.\nConsider a subset $F\\subset BS(1,2)$ and let $\\widetilde{F}$ be the associated subgraph.\nThe set of edges $\\overrightarrow{(f,g)}\\in E(\\widetilde{F})$ such that $\\overrightarrow{(g,f)}\\notin E(\\widetilde{F})$ we will denote $L(\\widetilde{F})$ (remark that unlike Section~\\ref{mainsection}, this is a set of edges rather than vertices).\nLet $\\overline{\\Gamma'}$ be the graph obtained by replacing every couple of edges $\\overrightarrow{(f,g)}$ and $\\overrightarrow{(g,f)}$ in $\\Gamma'$ with one undirected edge.\nLet $\\overline{\\widetilde{F}}$ be the subgraph of $\\overline{\\Gamma'}$ obtained by taking the images of $E(\\widetilde{F})\\setminus L(\\widetilde{F})$ and all adjacent vertices.\n\nFor any $i\\in\\Z$, consider the transformation $x\\mapsto x+2^i$ which is well defined on the vertices of $\\overline{\\Gamma'}$.\nDenote by $o(\\overline{\\widetilde{F}})$ the total amount of distinct non-trivial orbits in $V(\\overline{\\widetilde{F}})$ over all $i$.\nRemark that this quantity is well defined for any subgraph $\\overline{\\widetilde{F}}$ of $\\overline{\\Gamma'}$.\n\n\\begin{lemma}\\label{comparebs}\nConsider a subset $F\\subset BS(1,2)$ with $\\frac{|\\partial_{edge}F|}{|F|}\\leq1$.\nThen\n\n$$|F|\\geq|E(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})$$\nand\n$$\\frac{|\\partial_{edge}F|}{|F|}\\geq2\\frac{|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}{|E(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe start by estimating $|F|$.\nBy definition of associated subgraph, we have\n$$2|F|=|E(\\widetilde{F})|.$$\nBy definition of $L(\\widetilde{F})$ and $\\overline{\\widetilde{F}}$ we have\n$$|E(\\widetilde{F})=|L(\\widetilde{F})|+2|E(\\overline{\\widetilde{F}})|.$$\nWe now need to prove that $|L(\\widetilde{F})|\\geq2o(\\overline{\\widetilde{F}})$.\nIndeed, consider any non-trivial orbit and let $f$ and $g$ be respectively its smallest and largest elements.\nIf the step of the orbit is $2^i$, we have that $(i,f)$ and $(i,g)$ are elements of $F$.\nThen $\\overrightarrow{(f-2^i,f)}$ and $\\overrightarrow{(g,g+2^i)}$ must both be edges in $\\widetilde{F}$.\nAs $f-2^i$ and $g+2^i$ are not in $\\overline{\\widetilde{F}}$, those two edges are then in $L(\\widetilde{F})$.\n\nWe now estimate the boundary.\nWe will denote by $\\partial_{edge}^aF$ and $\\partial_{edge}^bF$ the edges in $\\partial_{edge}F$ labeled $a$ and $b$ respectively.\nIt follows directly from the definition of associated graph and $L(\\widetilde{F})$ that\n$$|\\partial_{edge}^bF|=|L(\\widetilde{F})|.$$\n\nConsider a vertex $f\\in V(\\overline{\\widetilde{F}})$.\nBy definition of $\\overline{\\widetilde{F}}$, there is $g$ such that $\\overrightarrow{(f,g)}\\in E(\\widetilde{F})$.\nIt follows that there is $x$ such that $(x,f)\\in F$.\nThere are thus at least two elements of $\\partial_{edge}^aF$ such that their corresponding configuration is $f$.\nWe obtain:\n$$|\\partial_{edge}^aF|\\geq2|V(\\overline{\\widetilde{F}})|.$$\nThen:\n$$\\frac{|\\partial_{edge}F|}{|F|}\\geq2\\frac{|V(\\overline{\\widetilde{F}})|+\\frac{1}{2}|L(\\widetilde{F})|}{|E(\\overline{\\widetilde{F}})|+\\frac{1}{2}|L(\\widetilde{F})|}\\geq2\\frac{|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}{|E(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}.$$\n\\end{proof}\n\nWe now seek to understand subgraphs of $\\overline{\\Gamma'}$ that optimize this quotient.\nWe claim that the subgraphs with vertices $[\\![1,n]\\!]$ do so.\nWe will denote $e(n)$ (respectively $o(n)$) the amount of edges (respectively orbits) in the subgraph with vertices $[\\![1,n]\\!]$ and all induced edges.\nWe prove results similar to Theorem~\\ref{thmharper}, which we cited from Haprer's book~\\cite{Harper2004}.\n\n\\begin{lemma}\\label{baums}\nLet $G$ be a subgraph of $\\overline{\\Gamma'}$ with $n$ vertices.\nThen\n\n$$|E(G)|\\leq e(n).$$\n\\end{lemma}\n\n\\begin{proof}\nWe start by estimating $e(n)$.\nFor $n\\in\\N$ with $2^{k-1}<n\\leq2^k$ we have that $e(n)$ is the sum over $i$ of the amount of couples of elements of $[\\![1,n]\\!]$ with difference $2^i$.\nIn other words,\n$$e(n)=\\sum_{i=0}^{k-1}(n-2^i)=kn-2^k+1.$$\n\nWe will now prove the result of the lemma by induction on $n$.\nThe base $n=1$ is trivial.\nFix a subgraph $G$ with $n$ vertices.\nUp to translation by an integer we can assume that the vertices of $G$ are integer and the smallest is $1$.\n\nAssume first that all vertices of $G$ are odd integers.\nLet $i$ be the largest integer such that $2^i$ divides all elements of $\\{f-1:f\\in V(G)\\}$.\nDefine $\\psi(f)=\\frac{f-1}{2^i}+1$.\nThen the set $\\psi(V(G))$ is a set of integers with the smallest element being $1$.\nFurthermore, for any $f$ and $f'$ divisible by $2^i$, $|f-f'|$ is a power of $2$ if and only if $|\\psi(f)-\\psi(f')|$ is.\nWithout loss of generality we can replace $G$ by the subgraph of $\\overline{\\Gamma'}$ with vertices $\\psi(V(G))$.\n\nWe can therefore assume $G$ has both even and odd elements.\nLet $G_1$ be the subgraph induced by vertices of $G$ that are odd integers, and $G_2$ the subgraph induced by even integers.\nDenote the cardinal of their vertex sets by $n_1$ and $n_2$ respectively.\nNotice that $n=n_1+n_2$.\n\nAs the difference (in terms of integer values) between vertices of $G_1$ and $G_2$ is always odd, there can be an edge if and only if the difference is $1$.\nThen by induction hypothesis the number of edges in $F$ is not greater than\n$$e(n_1)+e(n_2)+2\\min(n_1,n_2)-\\varepsilon$$\nwhere $\\varepsilon=1$ if $n_1$ and $n_2$ are equal, and $0$ otherwise.\n\nLet us denote $2^{k_1-1}<n_1\\leq2^{k_1}$, $2^{k_2-1}<n_2\\leq2^{k_2}$.\nWithout loss of generality, assume $n_1\\leq n_2$.\nThen $k-1\\leq k_2\\leq k$.\nLet $\\delta=k-k_2$.\nWe calculate\n\n\\begin{align*}\n& |E(G)|-e(n)\\leq e(n_1)+e(n_2)+2n_1-\\varepsilon-e(n) \\\\\n& =k_1n_1-2^{k_1}+1+k_2n_2-2^{k_2}+1+2n_1-\\varepsilon-k(n_1+n_2)+2^k-1 \\\\\n& =(k_1-k+2)n_1-\\delta n_2+1+2^k-2^{k_1}-2^{k_2}-\\varepsilon=A.\n\\end{align*}\n\nWe seek to prove that $A\\leq0$.\nWe have $\\delta=0$ or $1$.\nFirst, assume $\\delta=0$.\nThen $2^k=2^{k_2}$ and\n$$A=(k_1+2-k)n_1+1-2^{k_1}-\\varepsilon.$$\nAs $k_1\\leq k-1$, we have $A\\leq n_1+1-2^{k_1}-\\varepsilon\\leq 0$.\n\nAssume now $\\delta=1$.\nThen\n$$A=(k_1+2-k)n_1-n_2+1+2^{k-1}-2^{k_1}-\\varepsilon.$$\n\nAssume first that $k_1\\leq k-2$.\nThen $A\\leq 2^{k_2}-n_2+1-\\varepsilon\\leq0$.\n\nAs $k_1\\leq k_2=k-1$, the only case left is $k_1=k-1$.\nThen\n$$A=n_1-n_2+1-\\varepsilon.$$\nIf $n_1-n_2=0$, then $\\varepsilon=1$ and $A=0$.\nIf $n_1-n_2\\leq-1$, then $A\\leq-\\varepsilon\\leq0$.\n\\end{proof}\n\n\\begin{lemma}\\label{baums2}\nLet $G$ be a subgraph of $\\overline{\\Gamma'}$ such that $|V(G)|+o(G)=2n-1$ or $2n$.\nThen\n\n$$|E(G)|\\leq e(n).$$\n\\end{lemma}\n\nRemark that $o(n)=n-1$.\nIndeed, for each non-trivial orbit, consider the second smallest (in terms of the natural order on the integers) element of the orbit.\nIf it has step $2^i$, that element is between $1+2^i$ and $2^{i+1}$.\nTherefore these elements are disjoint, and cover every element in $[\\![1,n]\\!]$ except the element $1$.\n\n\\begin{proof}\nWe will now prove the result of the lemma by induction on $n$.\nThe base $n=1$ is trivial.\nFix a subgraph $G$ such that $|V(G)|+o(F)=2n-1$ or $2n$.\nUp to translation by an integer we can assume that the vertices of $G$ are integer and the smallest is $1$.\nAs in Lemma~\\ref{baums}, we can assume that $G$ has even vertices, and split it into $G_1$ and $G_2$ based on the parity of the vertices.\n\nLet $n_1=\\lceil\\frac{|V(G_1)|+o(G_1)}{2}\\rceil$ and $n_2=\\lceil\\frac{|V(G_2)|+o(G_2)}{2}\\rceil$.\nLet $2^{k_1-1}<n_1\\leq2^k_1$ and $2^{k_2-1}<n_2\\leq2^k_2$.\nWithout loss of generality, let $n_1\\leq n_2$.\n\nConsider first the case where there is at least one edge between $G_1$ and $G_2$.\nThen $o(G)\\geq o(G_1)+o(G_2)+1$ and $2n-1\\geq|V(G)|+o(G)-1\\geq|V(G_1)|+o(G_1)+|V(G_2)|+o(G_2)$.\nThis implies that $n_1+n_2\\leq n$.\nThe result then follows from the proof of Lemma~\\ref{baums}.\n\nConsider now the case where there is no edge between $G_1$ and $G_2$.\nThen $o(G)\\geq o(G_1)+o(G_2)$ and $2n\\geq|V(G)|+o(G)\\geq|V(G_1)|+o(G_1)+|V(G_2)|+o(G_2)$.\nThis implies that $n_1+n_2\\leq n+1$.\nBy induction hypothesis we have\n$$|E(G)|=|E(G_1)|+|E(G_2)|\\leq e(n_1)+e(n_2).$$\nWe calculate\n$$e(n_1)+e(n_2)-e(n)=k_1n_1+k_2n_2-kn+2^k-2^{k_1}-2^{k_2}+1=A.$$\nAssume first that $k_2=k$.\nThen\n$$A\\leq(k_1-k)n_1-2^{k_1}+1+k=(k_1-k)(n_1-1)-2^{k_1}+k_1+1\\leq0$$\nas $n_1\\geq1$.\n\nAssume now that $k_2\\leq k-2$.\nHowever, in that case $n_1+n_2\\leq2^{k-1}<n$, and the result follows from the proof of Lemma~\\ref{baums}.\n\nAssume that $k_2=k-1$.\nThen\n\\begin{equation*}\n\\begin{split}\nA&=(k_1+1-k)n_1-n+2^{k-1}-2^{k_1}+k\\\\\n&=(k_1+1-k)(n_1-1)-(n-2^{k-1})-(2^{k_1}-k_1-1)\\leq0,\n\\end{split}\n\\end{equation*}\nas $n_1\\geq1$ and $k_1\\leq k_2=k-1$.\n\\end{proof}\n\nCombining those two lemma, we obtain:\n\n\\begin{cor}\nLet $G$ be a subgraph of $\\overline{\\Gamma'}$ such that $|V(G)|+o(G)\\leq2n$ and $\\frac{|V(G)|+o(G)}{|E(G)|+o(G)}\\leq1$.\nThen\n\n$$\\frac{|V(G)|+o(G)}{|E(G)|+o(G)}\\geq\\frac{n+o(n)}{e(n)+o(n)}.$$\n\\end{cor}\n\nRemark that $n+o(n)=2n-1$.\n\n\\begin{proof}\nIf $o(G)\\leq o(n)$, the result follows trivially from Lemma~\\ref{baums2}.\nAssume then that $o(G)>o(n)=n-1$.\n\nThus $|V(G)|=n'\\leq n$.\nFrom Lemma~\\ref{baums} we obtain $|E(G)|\\leq e(n')$.\nWe have:\n\n$$\\frac{|V(G)|+o(G)}{|E(G)|+o(G)}\\geq\\frac{n'+o(n')+(o(G)-o(n'))}{e(n')+o(n')+(o(G)-o(n'))}\\geq\\frac{n'+o(n')}{e(n')+o(n')}\\geq\\frac{n+o(n)}{e(n)+o(n)}.$$\n\\end{proof}\n\nThe large inequality of Theorem~\\ref{bsthm} follows directly from this corollary and Lemma~\\ref{comparebs}.\nThe strict inequality is obtained by noticing that if $|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})<n+o(n)$, then $|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})\\leq2n-2$ and we can apply the corollary for $n-1$.\nThis concludes the proof of Theorem~\\ref{bsthm}.\n\nFinally, we consider $BS(1,p)$ with the standard generating set.\n\\begin{ex}\\label{exbsp}\nThere exist natural numbers $p$ and $n$ and a set $F\\subset BS(1,p)$ such that $|F|\\leq|F_n|$ (where $F_n=\\{(k,f):k\\in[\\![0,n-1]\\!],f\\in\\Z,0\\leq f<p^n\\}$) and\n\n$$\\frac{|\\partial_{edge}F|}{|F|}<\\frac{|\\partial_{edge}F_n|}{|F_n|}\\leq1.$$\nIn particular, this is true for $n=3$ and $p$ divisible by $3$ with $36\\leq p\\leq60$.\n\\end{ex}\n\n\\begin{proof}\nThe standard F{\\o}lner set $F_n$ has $np^n$ elements.\nWe calculate $|\\partial_{edge}^aF_n|=2p^n$ and $|\\partial_{edge}^bF_n|=2\\frac{p^n-1}{p-1}$.\nThen\n$$|\\partial_{edge}F_n|=2p^n\\frac{p-\\frac{1}{p^n}}{p-1}\\leq2p^n\\frac{p}{p-1},$$\nand $\\frac{|\\partial_{edge}F_n|}{|F_n|}\\leq\\frac{2}{n}\\frac{p}{p-1}$.\nTherefore for $n=3$ and $p\\geq3$ we have $\\frac{|\\partial_{edge}F_3|}{|F_3|}\\leq1$.\nAssume that $p$ is divisible by $3$ and let\n$$F=\\{(k,f):k\\in[\\![0,3]\\!],f=\\sum_{i=0}^3\\varepsilon_ip^i\\mbox{ with }0\\leq\\varepsilon_i<\\frac{p}{3}\\}.$$\nThen $|F|=4(\\frac{p}{3})^4=3p^3\\frac{4p}{243}$ and for $p\\leq60$ we obtain $|F|\\leq|F_3|$.\nFurthermore, $|\\partial_{edge}^aF|=2(\\frac{p}{3})^4$ and $|\\partial_{edge}^bF|=8(\\frac{p}{3})^3$.\nThus\n$$|\\partial_{edge}F|=4\\left(\\frac{p}{3}\\right)^4\\left(\\frac{1}{2}+\\frac{6}{p}\\right),$$\nand for $p\\geq36$ we have:\n$$\\frac{|\\partial_{edge}F|}{|F|}=\\frac{1}{2}+\\frac{6}{p}\\leq\\frac{2}{3}<\\frac{2}{3}\\frac{p-\\frac{1}{p^3}}{p-1}=\\frac{|\\partial_{edge}F_3|}{|F_3|}.$$\n\\end{proof}\nTherefore the result of Theorem~\\ref{bsthm} is not true for $BS(1,p)$ for all $p$.\n\n\\bibliographystyle{plainurl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \\label{sec:intro}\nThe reionization of the universe is the process where neutral hydrogen (H$_{I}$) becomes \nionized and determines the transition from an opaque state to the transparent intergalactic \nmedium (IGM) we observe today. Although the duration of this phase seems to be well \nestablished \\citep{Fan06,Bec15,Planck}, the source population causing this effect \nis still elusive. The  debate is still ongoing as to whether the main contributors \nare faint, high-redshift, star-forming galaxies (SFGs) or active galactic nuclei (AGNs).\nFor both populations there are critical issues. The assumption for the galaxies being \nmain contributors is that all faint SFGs should present an escape fraction of \nionizing radiation {\\it $f_{esc}$} of 10-20\\% \\citep{Fin15,Bow16}, which has not \nbeen observed so far, apart from in a handful of sources \\citep{Sha16,Van16,Bian17,Ste18}.\nRecent results \\citep{Fle18,Jon18,Nai18,Tan18} do not provide a clear evidence, but \nindicate that it is difficult for the global SFG population to reach the 10-20\\% level\nof escape fraction required to drive the reionization process. The main objection against \nAGNs being the main contributors is that the number of bright quasars at $z>$4 is not \nenough \\citep{Fan06,Cow09,HM12} and the number of faint AGNs at high redshifts is \nstill not well constrained. Based on X-ray samples, at low luminosities in this redshift \nrange, the space density of obscured AGNs is at least two times higher than the unobscured \npopulation \\citep{Mar16b}, indicating that optically selected luminosity functions (LFs) \ncould only be a lower limit.     \n\nIn order to answer the question of whether faint AGNs can contribute to the ionizing \nultraviolet background (UVB), three aspects need to be quantified: (i) the exact level \nof the UVB; (ii) the fraction of ionizing radiation escaping these sources \n({\\it $f_{esc}$}); and (iii) the faint slope of the AGN LF. \n\nObservations of the ionizing UVB intensity in the redshift range 2$<z<$5 and the global \nemissivity of ionizing photons indicate a relatively flat hydrogen photoionization \nrate \\citep{BeB13}, but not spatially uniform \\citep{Bos18}. Such large opacity \nfluctuations cannot be easily explained by low clustering populations like ultra-faint \ngalaxies and could require the existence of rare bright sources at high redshift \n\\citep{Bec15,Bec18,Cha15,Cha17}. As models start including larger AGN contributions, \npredicted temperatures are in agreement with observational constraints at $z\\sim$4-6 \n\\citep{Kea18} (but see \\citet{Puc18} for a different interpretation). \n\nThus, a sizable population of faint (L$\\sim$L$^{*}$) AGNs could contribute significantly \nto the UVB, as long as enough H$_{I}$ ionizing photons manage to escape the host galaxy \n\\citep{MH15,Kha16}. In this respect, a recent study by \\citet{Gra18}, using deep optical\/UV \nspectroscopy, found that faint AGNs (L$\\sim$L$^{*}$) at 3.6$<z<$4.2 present high escape \nfractions of ionizing radiation, with a mean value of 74\\%. This is in agreement with \nsimilar studies of bright quasars (M$_{1450}\\leq$ -26) at the same redshift range \n\\citep{Cri16}, and there is no indication of dependence on absolute luminosities. This \nmeans that, if such results are extrapolated to higher redshift (5$<z<$7), the AGN \ncontribution to the cosmic reionization process can become significant. At this point, \nknowledge of the exact number of faint AGNs at redshifts $z>4$ becomes crucial in \naccurately determining the level of this contribution.\n\nCurrently the consensus is that the LF for bright AGNs is well constrained, showing a \npeak at $z\\sim$3 and then rapidly declining \\citep{Bon07,Cro09}. However, for $z>$3 the \ndebate is still open, with various studies presenting contradicting results. Works \npresented by \\citet{Ike11} and \\citet{Gli11} suggest that the number of faint AGNs \nat $z>$3 is higher than expected, producing a steeper slope at the faint end of the LF. \nBut although the faint-end slopes are similar, the normalization factor $\\Phi^{*}$ derived \nby \\citet{Gli11} is three times higher than what calculated by \\citet{Ike11} and \nsubsequently reproduced by other studies \\citep[i.e][]{Mas12,Aki18}. These latter \nstudies report a strong decline in AGN numbers going from z=3 to z=4. In other words, \nthere is still wide disagreement on the actual shape and normalization of the \nLF at $z\\sim$4.\n\nWork by \\citet{Gia15}, including photometric and spectroscopic redshifts of X-ray-selected \nAGN candidates in the CANDELS GOODS-South region, has shown that at $z>$4 the probed AGN \npopulation could produce the necessary ionization rate to keep the IGM highly ionized \n\\citep{MH15}. This result is still controversial, with recent works claiming the opposite \n\\citep[i.e.][]{DAl17,Ric17,Aki18,Has18,Par18}. In fact, so far, the optical LFs at this \nredshift range and luminosities are based on a handful of spectroscopically confirmed \nsources (e.g., eight for \\citet{Ike11} and five for \\citet{Gia15}). Since the bulk of \nionizing photons come from AGNs close to L$^{*}$, it is mandatory to measure their LF \nat $z>$4 in this luminosity range. For this reason, we started a pilot study in the \nCOSMOS field, ideal for this kind of analysis thanks to its multi-wavelength catalog, \nX-ray, and radio coverage, which allows us to robustly select our AGN candidates. Here \nwe present the bright part of our spectroscopically confirmed sample of \nintermediate-\/low-luminosity AGNs, reaching an absolute magnitude of M$_{1450}$=-23 \nand discuss a robust determination of the space density at $z\\sim$4.\n\nThroughout the paper we adopt the $\\Lambda$ cold dark matter ($\\Lambda$CMD) concordance \ncosmological model (H$_{0}$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\\Omega_{M}$ = 0.3, and \n$\\Omega_{\\Lambda}$ = 0.7). All magnitudes are in the AB system.\n\n\\section{AGN Candidate Selection} \nThe selection of our sample is based on: (i) photometric redshifts, (ii) color-color \nselection, and (iii) X-ray emission. \n\n\\begin{deluxetable*}{ccccccccc}\n\\tablecaption{Color-Color Candidates \\label{tab:colorsel}}\n\\tablehead{\n\\colhead{ID} & \\colhead{R.A.} &\n\\colhead{Decl.} & \\colhead{i$_{AB}$} & \\colhead{$z_{phot}$} &\n\\colhead{$z_{spec}$} & \\colhead{(B$_{J}$-V$_{J}$)} &\n\\colhead{(r-i)} & \\colhead{X-ray}\\\\}\n\\startdata\n658294\\tablenotemark{*} & 149.467350 &1.855592 &21.056&-1.000 & 4.174 &1.40&0.25 & no \\\\\n1856470\\tablenotemark{*}& 150.475680 &2.798362 &21.282&0.000 & 4.110 &1.42&0.32 & yes \\\\\n1581239& 150.746170 &2.674495 &21.556&0.293 & -1.000&1.77&0.48 & no \\\\\n507779& 150.485630 &1.871927 &22.034&0.605 & 4.450 &4.94&0.55 & yes \\\\\n38736\\tablenotemark{*}& 150.732540 &1.516127 &22.088&-1.000 & 4.183 &1.69&0.64 & no \\\\\n1226535& 150.100980 &2.419435 &22.325&0.480 & 4.637 &1.68&0.43 & yes \\\\\n422327 & 149.701500 &1.638375 &22.409&0.343 & 3.201 &1.54&0.14 & no \\\\\n664641\\tablenotemark{*} & 149.533720 &1.809260 &22.436&0.338 & 3.986 &1.69&0.30 & no \\\\\n1163086\\tablenotemark{*}& 150.703770 &2.370019 &22.444&-1.000 & 3.748 &1.44&0.25 & yes \\\\\n330806\\tablenotemark{*} & 150.107380 &1.759201 &22.555&3.848 & 4.140 &1.48&0.30 & yes \\\\\n344777 & 150.188180 &1.664540 &22.634&0.392 & -1.000&1.89&-0.44 & no \\\\\n1450499& 150.115830 &2.563627 &22.685&0.280 & 3.355 &1.94&0.63 & no \\\\\n1687778& 150.006940 &2.779943 &22.715&0.437 & -1.000&1.96&0.44 & no \\\\\n96886  & 150.289380 &1.559480 &22.765&3.860 & -1.000&1.77&0.27 & no \\\\\n1573716& 150.729200 &2.739130 &22.783&0.376 & -1.000&1.35&0.48 & no \\\\\n346317 & 150.205950 &1.654837 &22.800&0.352 & -1.000&1.450&-0.21 & no \\\\\n1257518& 150.025190 &2.371214 &22.810&0.241 & -1.000&1.60&0.34 & no \\\\\n1322738& 149.444050 &2.424602 &22.839&0.428 & -1.000&1.92&0.71 & no \\\\\n1663056& 150.185000 &2.779340 &22.862&3.658 & -1.000&2.29&0.52 & no \\\\\n1719143& 149.755390 &2.738555 &22.873&-1.000 & 3.535 &1.76&0.23 & yes \\\\\n125420 & 150.222680 &1.510574 &22.898&0.181 & -1.000&1.83&0.53 & no \\\\\n867305 & 149.446230 &2.115336 &22.950&0.651 & -1.000&2.11&0.71 & no \\\\\n612661 & 149.838500 &1.829048 &23.011&4.229 & 4.351 &1.93&0.60  & no \\\\\n\\enddata\n\\tablenotetext{*}{Used for the LF}\n\\end{deluxetable*}\n\nWe use the photometric catalog and redshifts presented by \\citet{Ilb09}. This is a 30-band \ncatalog, spanning from NUV photometry to IRAC data, with calculated $z_{phot}$ in a \nregion covering 1.73 deg$^{2}$ in COSMOS. The reported $z_{phot}$ dispersion \nis $\\sigma_{(\\Delta z)\/(z_{s}+1)}$=0.007 at i$_{AB}<$22.5 and increases to \n$\\sigma_{(\\Delta z)\/(z_{s}+1)}$=0.012 at i$_{AB}<$24. As discussed in \\citet{Ilb09}, \ntheir $z_{phot}$ determination is mostly based on galaxy templates. However, as showed \nby \\citet{Gia15}, for z$>$4 the accuracy on the photometric redshift estimate is weakly \ndependent on the adopted spectral libraries but it is mainly driven by the Lyman break \nfeature at rest frame wavelength (912\\AA). To take into account possible larger errors \non photometric redshifts for the AGN population, we extended the redshift interval. Thus, \nwe obtained a list of 42 candidates that have a photometric redshift estimate in the \ninterval $3.0\\leq z_{phot}\\leq 5.0$ and a magnitude i$_{AB}<$23.0.\n\nTo increase our selection efficiency and mitigate shortcomings of the $z_{phot}$ technique, \nwe include a color criterion. Since we have a wide number of bands available, initially \nwe explored various combinations of color-color selections, i.e., (B$_{J}$-V$_{J}$) versus \n(r-i), (B$_{J}$-r) versus (r-i), (g-r) versus (r-i), (u$_{*}$-B$_{J}$) versus (r-i), and \n(u$_{*}$-g) versus (r-i). Cross-correlating those candidates with known AGNs from the \nliterature, and after exploratory spectroscopy with \nLDSS-3\\footnote{http:\/\/www.lco.cl\/telescopes-information\/magellan\/instruments\/ldss-3}, \nfor this pilot study we narrowed down our selection to the most promising criterion, \ni.e. (B$_{J}$-V$_{J}$) versus (r-i). In the (B$_{J}$-V$_{J}$) versus (r-i) color-color \ndiagram we consider as high-redshift AGN candidates the sources found in the locus \ndelimited by: \\\\\n\\\\\n(B$_{J}$-V$_{J}$)$>$1.3 \\\\\nand \\\\\n(r-i)$\\leq$0.60$\\times$(B$_{J}$-V$_{J}$) - 0.30.\\\\\n\\\\\nWith this criterion we obtained 23 candidates down to i$_{AB}$=23.0, summarized in \nTable \\ref{tab:colorsel}. We decided not to put any constraints on the morphology, \nsince the population of low-luminosity AGNs (M$_{1450}\\sim$-23) includes Seyferts, \nwhere the host galaxy could be visible. \n\n\\begin{figure}\n\\label{fig:spec}\n\\plotone{fig1.pdf}\n\\caption{The spectra of the six AGNs with $3.6\\leq z\\leq 4.2$ and i$_{AB}\\leq $23.0 \ndiscovered during our spectroscopic campaign with IMACS and LDSS-3. The red line corresponds \nto zero flux F$_{\\lambda}$, in arbitrary units.} \n\\end{figure}\n\nThere is a relatively small overlap between the candidates selected by the various methods. \nMore specifically, only 7\\% of the candidates selected by photometric redshifts are also \nincluded in the color-selected sample (three out of 42 objects), which is useful to \nincrease our completeness. This is a clear advantage with respect to the works of \n\\citet{Gli11} and \\citet{Ike11} which only used four bands for their selections.\n\nThe final criterion for the creation of our sample was X-ray emission. In practice, we \nselected 38 sources detected in X-rays by deep Chandra observations in the COSMOS field \n\\citep{Civ16} with $z_{phot}\\geq$3 and a limiting magnitude i$_{AB}<$23. These photometric \nredshifts were provided by \\citet{Mar16a} based on AGNs, galaxies or hybrid templates, \nas described in \\citet{Sal11}. This sample consists both of type-1 and type-2 AGNs, and \nrepresents an unbiased census of the faint AGN population at this redshift. Only eight of \nthe sources selected with the first two criteria present also emission in X-rays, while six \ncandidates have been selected both by X-ray and color criteria. \n\nOur final sample consists of 92 AGN candidates with magnitudes i$_{AB}<$23, that have been \nselected by at least one of the methods mentioned above. Thanks to extensive spectroscopic \ncampaigns carried out in the COSMOS field \\citep[e.g.][]{Bru09,Ike11,Civ12,Mar16a,Hasin18}, \n22 of our 92 candidates have secure spectroscopic redshifts. To establish the nature of \nthe remaining 70 sources (five of which have uncertain spectroscopic redshifts), we started \nan exploratory spectroscopic campaign at the Magellan Telescopes.  \n\n\\section{Spectroscopic Follow-up}  \n\nWe were awarded 2.5 nights with the wide-field Inamori-Magellan Areal Camera and \nSpectrograph \\citep[IMACS,][]{Dre11} on the 6.5m Magellan-Baade telescope at Las \nCampanas Observatory to obtain spectra for our AGN candidates. We observed a total of five \nmulti-slit masks with the IMACS f\/2 camera (27$\\arcmin$ diameter field of view) \nwith total exposure times ranging from 3hr to 6hr, during dark time in 2018 February and \nMarch. The width of the slits was 1$\\arcsec$.0 and the detector was used without binning \n(0$\\arcsec$.2\/pixel in the spatial direction). \n\nFor the three 6hr masks we used the 300 line mm$^{-1}$ red-blazed grism (300\\_26.7) with \nspectral sampling of 1.25{\\AA} pixel$^{-1}$, while for the two 3hr masks we used the 200 \nline mm$^{-1}$ grism that has a slightly lower resolution, sampling 2.04{\\AA} pixel$^{-1}$. \nIt is worth noting that the space density of our AGN candidates is such that only around \nthree objects typically fall in an IMACS field of view at this magnitude limit.\n\nWe observed a total of 16 AGN candidates with magnitudes ranging from i$_{AB}=$20 to 23.0, \nand for 14 of them we obtained robust redshift determination at $z>3$, resulting in an \nefficiency of $\\sim 88\\%$, and two uncertain redshifts at $z>3$. Out of the sub-sample with \nsecure redshifts, we found six AGNs in the redshift range $3.6\\leq z_{spec}\\leq 4.2$, and \neight AGNs with a measured redshift of either $3.1<z_{spec}<3.6$ or $4.2<z_{spec}<4.7$.\nThe masks were completed with less reliable AGN candidates at $z\\sim$4 obtained by relaxing \nthe criteria mentioned above and exploring different color selections with respect to the \n(B$_{J}$-V$_{J}$) versus (r-i) one. We also added, as fillers, fainter candidates, \ni$_{AB}\\leq$ 24.0, in order to explore the fainter regime of the AGN LF. \n\nAfter the spectroscopic campaign, 36 sources of the parent sample of 92 candidates had \nsecure spectroscopic redshifts. In addition, seven sources had uncertain spectroscopic \nclassification, either because of low signal-to-noise ratio or because only one line was \nvisible. This left 49 candidates with no redshift information. Figure 1 shows the spectra \nof the six AGNs with $3.6\\leq z_{spec}\\leq 4.2$ and i$_{AB}\\leq $23.0 discovered during \nour spectroscopic campaign with IMACS and LDSS-3. Most sources show characteristic AGN \nlines like Si$_{IV}$ and C$_{IV}$. The two sources that don not show Si$_{IV}$ and C$_{IV}$, \nhave been included for other strong high-ionization lines that are also characteristic of \nAGNs. More specifically, COSMOS664641 has strong O$_{VI}$ and N$_{V}$ lines, while the \nC$_{IV}$ line falls in a region of telluric absorption and this could be the reason it is \nnot observed. In the case of COSMOS247934 we also detect strong O$_{VI}$\/Ly$\\beta$, \nas well as He$_{II}$ lines, and it is relatively bright with $M_{1450}\\sim -23.2$. The fact \nthat these sources lack X-ray emission does not preclude their AGN nature, since there is \na number of examples of AGNs not detected by deep X-ray surveys \\citep{Ste02}. Moreover, \ntheir luminosities ($M_{1450}\\le -23.5$) are another indication of their nuclear activity. \nIn the subsequent analysis we only consider sources with secure spectroscopic redshifts \neither from our campaign or from the literature.\n\n\\begin{figure}\n\\label{fig:col}\n\\plotone{fig2.pdf}\n\\caption{Color-color diagram. Blue triangles show candidates selected by color. \nPentagons show candidates selected by photometric redshifts or X-ray emission and \nconfirmed by spectroscopy as AGNs. Green pentagons correspond to AGNs at 3.6$<z_{spec}<$4.2 \nand magenta pentagons show confirmed AGNs that have either z$_{spec}<$3.6 or z$_{spec}>$4.2. \nNotice that half of the confirmed AGNs are found outside the color selection locus and that \nhalf of the color selected candidates still need to be observed.} \n\\end{figure}\n\nThe distribution of our candidates in the color-color space can be seen in Figure 2. \nHere we plot the entire color-selected sample and indicate which sources were confirmed as \nAGNs, in the redshift range of interest, either after our spectroscopic campaign or from \nthe literature. We also indicate sources that lie in the color locus but their spectroscopic \nredshifts are either $z_{spec}<$3.6 or $z_{spec}>$4.2. A detailed presentation of the \nfull spectroscopic sample and a comprehensive description of the different color criteria \nare not the main aims of the present paper and will be discussed in a future work.\n\n\\begin{deluxetable*}{ccccccccc}\n\\tablecaption{Confirmed AGNs Used for Determining Space Density \\label{tab:spaced}}\n\\tablehead{\n\\colhead{ID} & \\colhead{R.A.} &\n\\colhead{Decl.} & \\colhead{$i_{AB}$} & \n\\colhead{$z_{spec}$} & \\colhead{$r_{AB}$} &\\colhead{M$_{1450}$} & References \\\\}\n\\startdata\n38736 & 150.732540 & 1.516127 & 22.088 & 4.183 & 22.897 & -23.341   & our spectroscopy \\\\\n247934 & 150.801300 & 1.657550 & 22.334 & 3.772 & 22.817 & -23.182  & our spectroscopy \\\\\n330806 & 150.107380 & 1.759201 & 22.555 & 4.140 & 23.105 & -23.110  & \\citet{Ike11} \\\\\n658294 & 149.467350 & 1.855592 & 21.056 & 4.174 & 21.603 & -24.630  & \\citet{Tru09} \\\\\n664641 & 149.533720 & 1.809260 & 22.436 & 3.986 & 22.946 & -23.182  & our spectroscopy \\\\\n899256 & 150.782210 & 2.285049 & 21.927 & 3.626 & 22.363 & -23.545  & our spectroscopy \\\\\n1054048\\tablenotemark{*} & 149.879200 & 2.225839 & 22.697 & 3.650 & 23.200 & -22.722  & \\citet{Mar16a} \\\\\n1159815 & 150.638440 & 2.391350 & 22.157 & 3.650 & 22.539 & -23.383  & \\citet{Ike11} \\\\\n1163086 & 150.703770 & 2.370019 & 22.444 & 3.748 & 22.863 & -23.122  & \\citet{Mar16a} \\\\\n1208399 & 150.259540 & 2.376141 & 21.424 & 3.717 & 21.488 & -24.478  & \\citet{Mar16a} \\\\\n1224733 & 150.208990 & 2.438466 & 21.147 & 3.715 & 21.485 & -24.480  & \\citet{Mar16a} \\\\\n1273346\\tablenotemark{*} & 149.776910 & 2.444306 & 22.779 & 4.170 & 23.274 & -22.952  & \\citet{Mar16a} \\\\\n1730531\\tablenotemark{*} & 149.843220 & 2.659095 & 22.900 & 3.748 & 23.439 & -22.545  & our spectroscopy \\\\\n1856470 & 150.475680 & 2.798362 & 21.282 & 4.110 & 21.753 & -24.445  & \\citet{Mar16a} \\\\\n1938843 & 149.845860 & 2.860459 & 22.160 & 3.630 & 22.619 & -23.290  & our spectroscopy \\\\\n1971812 & 149.472870 & 2.793400 & 21.887 & 3.610 & 22.179 & -23.717  & \\citet{Mar16a} \\\\\n\\enddata\n\\tablenotetext{*}{Not included in the space density bins because M$_{1450}>$-23.0}\n\\end{deluxetable*}\n\n\\section{Space Density Determination}\n\nThe advantage of doing this study in the COSMOS field is that it already contains extensive \nspectroscopic follow-up and extensive multi-wavelength data from radio to X-rays. Thus, \ncombining the confirmed candidates presented above, with known AGNs from the literature, \nwe obtain a sample of 16 spectroscopically confirmed AGNs with 3.6$<z<$4.2 and i$_{AB}<$23, \npresented in Table \\ref{tab:spaced}, along with the relative r$_{AB}$ for each source. The \ni$_{AB}$ magnitudes are from HST F814W band, while the r$_{AB}$ magnitudes are from the \nSubaru telescope and they are described in \\citet{Ilb09}.\n\nThe absolute magnitude at 1450 {\\AA} rest frame (M$_{1450}$) for each source has been \nderived from the $r_{AB}$ magnitude applying a K-correction according to the following \nformula:\n\\begin{equation}\nM_{1450} = r_{AB} - 2.5log(1+z_{spec})+K_{corr} \n\\end{equation}\nwhere\n\\begin{equation}\nK_{corr}=2.5\\alpha_\\nu log_{10}(\\lambda_{obs}\/(1+z_{spec})\/\\lambda_{rest})\n\\end{equation}\nThe AGN intrinsic slope $\\alpha_\\nu$ is fixed to -0.7, while $\\lambda_{obs}=6284$ {\\AA} \nis the central wavelength of the $r_{AB}$ filter and $\\lambda_{rest}=1450$ {\\AA}. The reason \nwe chose the $r_{AB}$ band is because it is not affected by strong quasar emission lines, \nlike C$_{IV}$ that falls in the $i_{AB}$ band for this redshift range. The K-correction \nis redshift dependent and ranges from 0.05mag at z=3.6 to 0.14mag at z=4.2. To check the \nrobustness of our absolute magnitude determination, we have used various methods to \ncalculate it (using the i and r band, with and without K-correction). The point at \nM$_{1450}$=-24.5 remains basically unaltered, and we only see small changes in it at \nM$_{1450}$=-23.5, with the current determination resulting in the faintest absolute \nmagnitudes, thus being the most conservative.\n\nWe find four sources for -25$<M_{1450}<$-24 and 9 for -24$<M_{1450}<$-23. Based on these \nnumbers, we calculate the space density of AGNs at this redshift range in the two magnitude \nbins. This space density, summarized in Table \\ref{tab:spacedn}, has been derived by \ndividing the actual number of the spectroscopically confirmed AGNs with the comoving \nvolume between $3.6\\leq z\\leq 4.2$, without any correction for incompleteness. Thus it \nrepresents a robust lower limit to the real space density of $z\\sim 4$ AGNs. As can be \nseen in Figure 3, without any corrections (blue filled squares), our measurements agree \nwell with the \\citet{Gli11} analysis and put more stringent constraints on the knee \nof the LF.\n\nConsidering completeness and contamination, we can estimate rough corrections. \nIn the color selection, we have six AGNs with 3.6$<z_{spec}<$4.2 out of 12 AGNs with \nknown $z_{spec}$. Thus, out of the 11 candidates without $z_{spec}$, we expect five or six \n($\\sim$50\\%) to have 3.6$<z_{spec}<$4.2, resulting to 11 or 12 potential AGNs with the \n(B$_{J}$-V$_{J}$) versus (r-i) selection. The known AGNs with 3.6$<z_{spec}<$4.2 and \ni$_{AB}\\leq$23.0 are 16, of which we recover 37.5\\% with the (B$_{J}$-V$_{J}$) versus \n(r-i) criterion (six out of 16). This means that the total number of AGNs expected in \nCOSMOS at 3.6$<z_{spec}<$4.2 and i$_{AB}<$23.0 could be N$_{tot}$=11$\\times$16\/6=29.3 or \nN$_{tot}$=12$\\times$16\/6=32. The space density determination using only the 16 AGNs known \nat 3.6$<z_{spec}<$4.2, i$_{AB}\\leq$23.0 and -26.0$<M_{1450}<$-23.0 is incomplete by a factor \nof 0.50-0.55 at least. If we correct for this incompleteness factor, we will go to a level \nhigher than \\cite{Gli11} (green open squares in Figure 3). Adopting a slightly different \ncolor criterion, with ($B_{J}-V_{J})>1.1$ instead of 1.3 as threshold, the expected total \nnumber of AGNs is 34 and the completeness corrections remain at the $\\sim 50\\%$ level. \nThis indicates that the green squares (corrected space density) in Figure 3 are quite \nrobust with respect to the details of the adopted color criterion.\n\n\\begin{deluxetable*}{cccccc}\n\\tablecaption{AGN Space Density ($<z>$=3.9)\\label{tab:spacedn}}\n\\tablehead{\n\\colhead{M$_{1450}$} & \\colhead{$\\Phi$} &\n\\colhead{$\\sigma_\\Phi^{up}$} & \\colhead{$\\sigma_\\Phi^{low}$} & \n\\colhead{N$_{AGN}$} & \\colhead{$\\Phi_{corr}$}\\\\ & $Mpc^{-3} Mag^{-1}$ & & & & \\\\}\n\\startdata\n-24.5 & 3.509e-07& 2.789e-07& 1.699e-07& 4 &7.018e-07\\\\\n-23.5 &7.895e-07& 3.616e-07& 2.595e-07& 9 & 1.579e-06 \\\\\n\\enddata\n\\end{deluxetable*}\n\nIn Figure 3 we also present the LFs calculated by \\citet{Aki18}, \\citet{Par18}, and \n\\citet{Mas12} for comparison. The sample created by \\citet{Aki18} is limited to g-band \ndropout (i.e. $3.5<z<4.0$) point-like sources, for which they have derived photometric \nredshifts based on five filters (g,r,i,z,y). The faint-end slope presented in \ntheir work is too shallow to be reconciled with our measurements, which correspond to \nspectroscopically confirmed sources. The LF by \\citet{Ike11} is also lower than our space \ndensity determination. On the other hand \\citet{Par18}, after performing an independent \nphotometric redshift estimate of the X-ray-selected sample presented by \\citet{Gia15}, \ndiscarded 10 of the 22 sources that were supposed to lie at $z>$4. Even though the faint-end \nslope in \\citet{Par18} is steeper than that found by \\citet{Gli11}, their space density in \nabsolute magnitudes M$_{1450}<$-23 is marginally in agreement with our estimates. We also \nshow the space density derived by \\citet{Mar16b}, based on X-ray data, after being converted \nto UV \\citep{Ric17}. Although these points are higher than most optical LFs, they are slightly \nlower than our estimate. In Table \\ref{tab:spacedn} we present the estimate of the AGN space \ndensity $\\Phi$, based on our analysis, in the two absolute magnitude bins. Even excluding \nthe COSMOS247934, which is the least certain among our sources, the uncorrected space density \nat M$_{1450}$=-23.5 becomes 7.018e-07 $Mpc^{-3}Mag^{-1}$, which is still higher than all LFs \npresented in Figure 3, except for \\citet{Par18}. Considering the space density corrected for\nincompleteness, also the \\citet{Par18} LF also turns out to be underestimated.\n\nAn important aspect, made clear by our sample, is that selections based on color criteria \ncan be highly incomplete, since out of the 16 spectroscopically confirmed AGNs only six \nhave been selected by color. So far, the majority of studies on the AGN LF at this redshift \nrange is based on color-selected samples and this could be the reason why faint AGN number \ndensities have been underestimated. When a first attempt was made by \\citet{Gia15} to create \nan AGN sample based on non-traditional criteria, a different picture emerged. Given that AGNs, \neven at faint magnitudes, have a large escape fraction as shown by \\citet{Gra18}, an increase \nof the estimate of their population can have significant implications on the contribution \nof AGNs to the H$_{I}$ ionizing background.  \n\n\\section{Discussion and Conclusions}\n\nOur estimates of the space density in the range $-24.5<M_{1450}<-23.5$, shown by filled \nblue squares in Figure 3, are not corrected for any incompleteness factor, thus they \nrepresent firm lower limits, assuming that the density fluctuations due to cosmic variance \nare not important.\n\nTo explore this possibility, we checked how the volume density of AGNs at $z\\sim$4 in COSMOS \ncompares to the average derived from the SDSS survey. In the COSMOS area there is no known \nquasi stellar object (QSO) brighter than i$_{AB}$=21.0 in the redshift interval $3.6<z<4.2$. \nConsidering the SDSS DR14 catalog \\citep{Pas18} in areas of different sizes, ranging from \n10-100 deg$^{2}$, centered on the COSMOS field, we find a mean density of $0.59\\pm 0.10~deg^{-2}$ \ncompared to the mean SDSS density of $0.61\\pm 0.01~deg^{-2}$ (5683 QSOs in 9376 deg$^{2}$).\nThis indicates that the COSMOS field is not particularly overdense or underdense at $z\\sim 4$. \nFor this reason, the completion of a spectroscopic AGN survey in this field, such as the \none presented here, is fundamental to address the role of AGNs in the reionization epoch. \n\nIndeed, the AGN space density derived by our study is a conservative estimate and might \nincrease in the future. As can be seen in Table \\ref{tab:colorsel}, six confirmed AGNs at \n$z_{spec}\\geq 3$ have $z_{phot}<$1 in \\cite{Ilb09}. In two cases, where both the spectroscopic \nand the photometric redshifts are larger than 3, the $z_{phot}$ tends to systematically \nunderestimate the actual $z_{spec}$. This is an indication that our selection based on \n$z_{phot}$ could still be biased against $z\\sim 4$ AGNs, and that different $z_{phot}$ recipes \ncould increase the number of AGN candidates selected by photometric redshifts. As an example, \n\\citet{Sal11} provided photometric redshifts only for two of our sources in \nTable \\ref{tab:colorsel}, i.e., source id=330806 with $z_{phot}=3.949$ and id=1226535 \nwith $z_{phot}=4.545$. Their estimates are in reasonable agreement with the spectroscopic \nredshifts of these AGNs.\n\n\\begin{figure*}\n\\label{fig:spDen}\n\\plotone{fig3.pdf}\n\\caption{Luminosity function of quasi stellar objects\/AGNs at $z\\sim$4. Black triangles \nshow the bright end determined by the SDSS \\citep{Aki18}, green triangles are the points \npresented by \\citet{Sch18}, red circles show the bins calculated by \\citet{Gli11}, \nblue stars show the bins by \\citet{Gia15}, filled blue squares represent the two magnitude \nbins from this work (no corrections), and the open green squares correspond to the space \ndensity derived in this work after applying the corrections discussed in Section 4. Notice \nthat our data actually compensate for the apparent dip present in the work by \\citet{Gli11}. \nThe results by \\citet{Mar16b}, converted into UV, are presented as orange asterisks. For \ncomparison we also present the LFs derived by \\citet{Ike11}, \\citet{Mas12}, \\citet{Ric17}, \n\\citet{Aki18}, and \\citet{Par18}. All LFs have been evolved to z=3.9 following the density \nevolution law by \\citet{Fan01}.}\n\\end{figure*}\n\n\\citet{Stev18}, in their analysis of the UV LF of a mixed sample including both SFGs and \ngalaxies dominated by AGNs at $z\\sim$4 in the SHELA survey, found a space density of \n$10^{-6} Mpc^{-3}$ at $M_{1450}=-23.5$. Our measurements in the same redshift and absolute \nmagnitude is $\\sim1.6\\times10^{-6} Mpc^{-3}$ (corrected), and this is an indication that \ntheir LF is dominated by AGNs at $M_{1450}<-23.5$. Based on their discussion, a high AGN \nspace density would mean that AGNs could be largely responsible for the H$_{I}$ ionizing \nUVB at $z=4$.\n\nWe are confident that on the net of all random and systematic effects, the corrected \nestimate of the LF presented in this work represents a robust determination of the space \ndensity of L$^{*}$ AGNs at $z\\sim 4$. The agreement of our measurement with the \\citet{Gli11} \nresults favors at $M_{1450}<-23.5$ a steeper slope of the LF for the COSMOS field than that \ndetermined by \\citet{Ike11}, \\citet{Mas12}, \\citet{Ric17}, \\citet{Aki18}, and \\citet{Par18}. \n\nThis study poses an open challenge that should be addressed in the future with major \nobservational effort: to measure with great accuracy the space density of AGNs at L=L$^{*}$ \nand $z>4$. In fact, in a subsequent work, once our spectroscopic sample is complete, we will \npresent the global shape of the LF at $z\\sim$4 and the associated emissivity. This can have \ndeep implications on the extrapolation of the number of QSOs expected at high-z in wide and \ndeep large area surveys, either ground based, e.g. LSST, or from space, e.g., e-Rosita, \nEuclid, WFIRST. An upward revision of the number density of L=L$^{*}$ AGNs would certainly \nimply a reconsideration of the expected QSO and AGN numbers at $z>4$ in these future missions. \n\n\\acknowledgments\nWe would like to thank the anonymous referee for useful suggestions and constructive \ncomments that helped us improve this paper. This paper includes data gathered with the \n6.5 meter Magellan Telescopes located at Las Campanas Observatory (LCO), Chile.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nWe consider the equation\n\\begin{equation}\n  \\label{eq:1}\n  i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + |\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n\\end{equation}\nand both semi-classical ($\\eps\\to 0$) and large time ($t\\to \\pm\n\\infty$) limits. Of course these limits must not be expected to\ncommute, and one of the goals of this paper is to analyze this lack of\ncommutation on specific asymptotic data, under the form of coherent\nstates as described below. Even though our main  result\n(Theorem~\\ref{theo:cv}) is proven specifically for the above case of a\ncubic three-dimensional equation, two important intermediate results\n(Theorems~\\ref{theo:scatt-quant} and \\ref{theo:scatt-class}) are\nestablished in a more general setting. Unless specified otherwise, we\nshall from now on consider $\\psi^\\eps:\\R_t \\times \\R^d_x\\to \\C$, $d\\ge\n1$. \n\n\n\\subsection{Propagation of initial coherent states}\n\\label{sec:prop-init-coher}\n\nIn this subsection, we consider the initial value problem, as opposed\nto the scattering problem treated throughout this paper. More\nprecisely, we assume here that the wave function is, at time $t=0$,\ngiven by the coherent state\n\\begin{equation}\n  \\label{eq:ci}\n  \\psi^\\eps(0,x) = \\frac{1}{\\eps^{d\/4}}a\\(\\frac{x-q_0}{\\sqrt\\eps}\\)\n  e^{ip_0\\cdot (x-q_0)\/\\eps},\n\\end{equation}\nwhere $q_0,p_0\\in \\R^d$ denote the initial position and velocity,\nrespectively. The function $a$ belongs to the Schwartz class,\ntypically. In the case where $a$ is a (complex) Gaussian, many\nexplicit computations are available in the linear case (see\n\\cite{Hag80}). Note that the $L^2$-norm of $\\psi^\\eps$ is independent\nof $\\eps$, $\\|\\psi^\\eps(t,\\cdot)\\|_{L^2(\\R^d)} =\\|a\\|_{L^2(\\R^d)}$. \n\n Throughout this subsection, we assume that the external\npotential $V$ is smooth and real-valued, $V\\in C^\\infty(\\R^d;\\R)$, and\nat most quadratic, in the sense that \n\\begin{equation*}\n  \\d^\\alpha V\\in L^\\infty(\\R^d),\\quad \\forall |\\alpha|\\ge 2.\n\\end{equation*}\nThis assumption will be strengthened when large time behavior is\nanalyzed. \n\\subsubsection{Linear case}\n\\label{sec:linear-case}\n Resume \\eqref{eq:1} in the absence of nonlinear term:\n\\begin{equation}\n  \\label{eq:lin}\n  i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps,\\quad x\\in \\R^d,\n\\end{equation}\nassociated with the initial datum \\eqref{eq:ci}. To derive an\napproximate solution, and to describe the propagation of the initial\nwave packet, introduce the Hamiltonian flow\n\\begin{equation}\n  \\label{eq:hamil}\n  \\dot q(t)= p(t),\\quad \\dot p(t)=-\\nabla V\\(q(t)\\),\n\\end{equation}\nand prescribe the initial data $q(0)=q_0$, $p(0)=p_0$. Since the\npotential $V$ is smooth and at most quadratic, the solution\n$(q(t),p(t))$ is smooth, defined for all time, and grows at most\nexponentially.\nThe classical action is given by\n\\begin{equation}\\label{eq:action}\nS(t)=\\int_0^t \\left( \\frac{1}{2} |p(s)|^2-V(q(s))\\right)\\,ds.\n\\end{equation}\nWe observe that if we change the unknown function $\\psi^\\eps$ to\n$u^\\eps$ by\n\\begin{equation}\n  \\label{eq:chginc}\n  \\psi^\\eps(t,x)=\\eps^{-d\/4} u^\\eps \n\\left(t,\\frac{x-q(t)}{\\sqrt\\eps}\\right)e^{i\\left(S(t)+p(t)\\cdot\n    (x-q(t))\\right)\/\\eps},\n\\end{equation}\nthen, in terms of $u^\\eps=u^\\eps(t,y)$, the Cauchy problem\n\\eqref{eq:lin}--\\eqref{eq:ci} is equivalent to \n\\begin{equation}\\label{eq:ueps0}\ni\\d_t\nu^\\eps+\\frac{1}{2}\\Delta u^\\eps=V^\\eps(t,y)\nu^\\eps\\quad ;\\quad u^\\eps(0,y) = a(y),\n\\end{equation}\nwhere the external time-dependent potential $V^\\eps$ is given by\n\\begin{equation}\n  \\label{eq:Veps}\n  V^\\eps(t,y)= \\frac{1}{\\eps}\\(V(x(t)+\n\\sqrt{\\eps}y)-V(x(t))-\\sqrt{\\eps}\\<\\nabla V(x(t)),y\\>\\).\n\\end{equation}\nThis potential corresponds to the first term of a Taylor expansion of\n$V$ about the point $q(t)$, and we naturally introduce \n$u=u(t,y)$ solution to \n\\begin{equation}\\label{eq:ulin}\ni\\d_tu+\\frac{1}{2}\\Delta u=\\frac{1}{2}\\< Q(t)y,y\\> u\\quad\n;\\quad u(0,y)=a(y),\n\\end{equation}\nwhere\n\\begin{equation*}\n  Q(t):= \\nabla^2 V\\(q(t)\\), \\quad \\text{so that } \\frac{1}{2}\\<\n  Q(t)y,y\\>  = \\lim_{\\eps \\to 0} V^\\eps(t,y). \n\\end{equation*}\nThe obvious candidate to approximate the initial wave function\n$\\psi^\\eps$ is then:\n\\begin{equation}\n  \\label{eq:phi}\n  \\varphi^\\eps(t,x)=\\eps^{-d\/4} u\n\\left(t,\\frac{x-q(t)}{\\sqrt\\eps}\\right)e^{i\\left(S(t)+p(t)\\cdot\n    (x-q(t))\\right)\/\\eps}.\n\\end{equation}\nIndeed, it can be proven (see\ne.g. \\cite{BGP99,BR02,CoRoBook,Hag80,HaJo00,HaJo01}) that there\nexists \n$C>0$ independent of $\\eps$ such that\n\\begin{equation*}\n  \\|\\psi^\\eps(t,\\cdot)-\\varphi^\\eps (t,\\cdot)\\|_{L^2(\\R^d)}\\le\n  C\\sqrt\\eps e^{Ct}. \n\\end{equation*}\nTherefore, $\\varphi^\\eps$ is a good approximation of $\\psi^\\eps$ at least up to time\nof order $c\\ln\\frac{1}{\\eps}$ (Ehrenfest time). \n\n\\subsubsection{Nonlinear case}\n\\label{sec:nonlinear}\n When adding a nonlinear term to \\eqref{eq:lin}, one has to be\n cautious about the size of the solution, which rules the importance\n of the nonlinear term. To simplify the discussions, we restrict our\n analysis to the case of a gauge invariant, defocusing, power nonlinearity,\n $|\\psi^\\eps|^{2\\si}\\psi^\\eps$. We choose to measure the importance of\n nonlinear effects not directly through the size of the initial data,\n but through an $\\eps$-dependent coupling factor: we keep the initial\n datum \\eqref{eq:ci} (with an $L^2$-norm independent of $\\eps$), and\n consider\n \\begin{equation*}\n   i\\eps\\d_t \\psi^\\eps + \\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n   V(x)\\psi^\\eps + \\eps^\\alpha|\\psi^\\eps|^{2\\si}\\psi^\\eps.\n \\end{equation*}\nSince the nonlinearity is homogeneous, this approach is equivalent to\nconsidering $\\alpha=0$, up to multiplying the initial datum by\n$\\eps^{\\alpha\/(2\\si)}$. \nWe assume $\\si>0$, with $\\si<2\/(d-2)$ if $d\\ge 3$: for $a\\in \\Sigma$,\ndefined by\n\\begin{equation*}\n  \\Sigma = \\{f\\in H^1(\\R^d),\\quad x\\mapsto \\<x\\> f(x)\\in\n  L^2(\\R^d)\\},\\quad \\<x\\>=\\(1+|x|^2\\)^{1\/2},\n\\end{equation*}\nwe have, for fixed $\\eps>0$, $\\psi^\\eps_{\\mid t=0}\\in \\Sigma$, and the\nCauchy problem is globally well-posed, $\\psi^\\eps\\in C(\\R_t;\\Sigma)$\n(see e.g. \\cite{Ca11}). It was established in \n\\cite{CaFe11} that the value \n\\begin{equation*}\n  \\alpha_c = 1+\\frac{d\\si}{2}\n\\end{equation*}\nis critical in terms of the effect of the nonlinearity in the\nsemi-classical limit $\\eps\\to 0$. If $\\alpha>\\alpha_c$, then \n$\\varphi_{\\rm lin}^\\eps$, given by \\eqref{eq:ulin}-\\eqref{eq:phi},  is\nstill a good approximation of $\\psi^\\eps$ at least up to time\nof order $c\\ln\\frac{1}{\\eps}$. On the other hand, if\n$\\alpha=\\alpha_c$, nonlinear effects alter the behavior of $\\psi^\\eps$\nat leading order, through its envelope only. Replacing \\eqref{eq:ulin}\nby \n\\begin{equation}\\label{eq:u}\ni\\d_tu+\\frac{1}{2}\\Delta u=\\frac{1}{2}\\< Q(t)y,y\\> u+|u|^{2\\si}u,\n\\end{equation}\nand keeping the relation \\eqref{eq:phi}, $\\varphi^\\eps$ is now a good\napproximation of $\\psi^\\eps$. In \\cite{CaFe11} though, the time of\nvalidity of the approximation is not always proven to be of order at\nleast $c\\ln\\frac{1}{\\eps}$, sometimes shorter time scales (of the\norder $c\\ln\\ln\\frac{1}{\\eps}$) have to be considered, most likely for\ntechnical reasons only. Some of these restrictions have been removed in\n\\cite{Ha13}, by considering decaying external\npotentials $V$. \n\n\n\\subsection{Linear scattering theory and coherent states}\n\\label{sec:line-scatt-theory}\n\nWe now consider the aspect of large time, and instead of prescribing\n$\\psi^\\eps$ at $t=0$ (or more generally at some finite time), we\nimpose its behavior at $t=-\\infty$.\n In the linear case \\eqref{eq:lin},\nthere are several results addressing the question mentioned above,\nconsidering different forms of asymptotic states at $t=-\\infty$.\nBefore describing them, we recall important facts concerning quantum\nand classical scattering. \n\n\\subsubsection{Quantum scattering}\n\\label{sec:quantum-scattering}\n\nThroughout this paper, we assume that the external potential is\nshort-range, and satisfies the following properties:\n\\begin{hyp}\\label{hyp:V}\n  We suppose that $V$ is smooth and real-valued, $V\\in\n  C^\\infty(\\R^d;\\R)$. In addition, it is short range in the following\n  sense: there exists $\\mu>1$ such that\n  \\begin{equation}\n    \\label{eq:short}\n    |\\d^\\alpha V(x)|\\le \\frac{C_\\alpha}{(1+|x|)^{\\mu+|\\alpha|}},\\quad\n   \\forall \\alpha\\in \\N^d. \n  \\end{equation}\n\\end{hyp}\nOur final result is established under the stronger condition\n$\\mu>2$ (a condition which is needed in several steps of the proof), but\nsome results are established under the mere assumption\n$\\mu>1$. Essentially, the analysis of the approximate solution is valid\nfor $\\mu>1$\n(see Section~\\ref{sec:class}), while the rest of the analysis requires $\\mu>2$. \n\\smallbreak\n\nDenote by \n\\begin{equation*}\nH_0^\\eps= -\\frac{\\eps^2}{2}\\Delta\\quad \\text{and}\\quad\nH^\\eps=-\\frac{\\eps^2}{2}\\Delta+V(x) \n\\end{equation*}\nthe underlying Hamiltonians. For fixed $\\eps>0$, the (linear) wave\noperators are given by\n\\begin{equation*}\n  W_\\pm^\\eps = \\lim_{t\\to \\pm \\infty}e^{i\\frac{t}{\\eps}H^\\eps}e^{-i\\frac{t}{\\eps}H^\\eps_0},\n\\end{equation*}\nand the (quantum) scattering operator is defined by\n\\begin{equation*}\n  S^\\eps_{\\rm lin} = \\(W_+^\\eps\\)^* W_-^\\eps. \n\\end{equation*}\nSee for instance \\cite{DG}.\n\n\\subsubsection{Classical scattering}\n\\label{sec:classical-scattering}\n\nLet $V$ satisfying Assumption~\\ref{hyp:V}. \nFor $(q^-,p^-)\\in \\R^d\\times \\R^d$, we consider the classical\ntrajectories  $(q(t),p(t))$ defined by \\eqref{eq:hamil}, \nalong with the prescribed asymptotic behavior as $t\\to -\\infty$:\n\\begin{equation}\n  \\label{eq:CI-hamilton}\n  \\lim_{t\\to -\\infty}\\left| q(t)-p^- t -q^-\\right| = \\lim_{t\\to\n    -\\infty} |p(t)-p^-|=0. \n\\end{equation}\nThe existence and uniqueness of such a trajectory can be found in\ne.g. \\cite{DG,ReedSimon3}, provided that $p^-\\not =0$. Moreover, there\nexists a closed set $\\mathcal N_0$ of Lebesgue measure zero in\n$\\R^{2d}$ such that for all $(q^-,p^-)\\in \\R^{2d}\\setminus \\mathcal\nN_0$, there exists $(q^+,p^+)\\in \\R^d\\times\n  \\(\\R^d\\setminus\\{0\\}\\)$ such that\n\\begin{equation*}\n  \\lim_{t\\to +\\infty}\\left| q(t)-p^+ t -q^+\\right| = \\lim_{t\\to\n    +\\infty} |p(t)-p^+|=0. \n\\end{equation*}\nThe classical scattering operator is $S^{\\rm cl}:(q^-,p^-)\\mapsto\n(q^+,p^+)$. Choosing  $(q^-,p^-)\\in \\R^{2d}\\setminus \\mathcal\nN_0$ implies that the following assumption is satisfied:\n\\begin{assumption}\\label{hyp:flot}\n  The asymptotic center in phase space, $(q^-,p^-)\\in \\R^d\\times\n  \\(\\R^d\\setminus\\{0\\}\\)$ is such that the classical scattering\n  operator is well-defined, \n  \\begin{equation*}\n    S^{\\rm cl}(q^-,p^-)=\n(q^+,p^+),\\quad p^+\\not =0,\n  \\end{equation*}\nand the classical action has  limits as $t\\to \\pm\\infty$:\n\\begin{equation*}\n  \\lim_{t\\to -\\infty}\\left|S(t)-t\\frac{|p^-|^2}{2}\\right| =\n  \\lim_{t\\to +\\infty}\\left|S(t)-t\\frac{|p^+|^2}{2}-S_+\\right|  =0,\n\\end{equation*}\nfor some $S_+\\in \\R$. \n\\end{assumption}\n \n\\subsubsection{Some previous results}\n\\label{sec:some-prev-results}\n\n\nIt seems that the first mathematical result involving both the\nsemi-classical and large time limits appears in\n\\cite{GV79mean}, where the classical field limit of non-relativistic\nmany-boson theories is studied in space dimension $d\\ge 3$. \n\nIn\n\\cite{Yajima79}, the\ncase of a short range potential (Assumption~\\ref{hyp:V}) is\nconsidered, with asymptotic states \nunder the form of semi-classically concentrated functions,\n\\begin{equation*}\n  e^{-i\\frac{\\eps t}{2}\\Delta}\\psi^\\eps(t)_{\\mid t =-\\infty}\n  =\\frac{1}{\\eps^{d\/2}}\\widehat f\\(\\frac{x-q^-}{\\eps}\\),\\quad f\\in L^2(\\R^d),\n\\end{equation*}\nwhere $\\widehat f$ denotes the standard Fourier transform (whose\ndefinition is independent of $\\eps$). The main result from\n\\cite{Yajima79} shows that the semi-classical limit for $S^\\eps_{\\rm lin}$\ncan be expressed in terms of the classical scattering operator, of the\nclassical action, and of\nthe Maslov index associated to each classical trajectory. We refer to\n\\cite{Yajima79} for a precise statement, and to \\cite{Yaj81} for the\ncase of long range potentials, requiring modifications of the\ndynamics. \n\\smallbreak\n\nIn \\cite{Hag81,HaJo00}, coherent  states are considered,\n\\begin{equation}\n  \\label{eq:asym-state}\n e^{-i\\frac{\\eps t}{2}\\Delta}\\psi^\\eps(t)_{\\mid t =-\\infty}=\n  \\frac{1}{\\eps^{d\/4}}u_-\\(\\frac{x-q^-}{\\sqrt\\eps}\\) \n  e^{ip^-\\cdot (x-q^-)\/\\eps+iq^-\\cdot p^-\/(2\\eps)}=:\\psi_-^\\eps(x).\n\\end{equation}\nMore precisely, in \\cite{Hag81,HaJo00}, the asymptotic state $u_-$ is assumed\nto be a complex Gaussian function. Introduce the notation\n\\begin{equation*}\n  \\delta(t) = S(t)-\\frac{q(t)\\cdot p(t) - q^-\\cdot p^-}{2}.\n\\end{equation*}\nThen Assumption~\\ref{hyp:flot} implies that there exists $\\delta^+\\in\n\\R$ such that\n\\begin{equation*}\n  \\delta(t)\\Tend t {-\\infty}0\\quad \\text{and}\\quad \\delta(t)\\Tend t\n  {+\\infty} \\delta^+. \n\\end{equation*}\n In \n\\cite{CoRoBook,HaJo00}, we find the following general result  (an asymptotic\nexpansion in powers of $\\sqrt\\eps$ is actually given, but we stick to\nthe first term to ease the presentation):\n\\begin{theorem}\\label{theo:version-lineaire}\n  Let Assumptions~\\ref{hyp:V} and \\ref{hyp:flot} be satisfied, and let\n  \\begin{equation*}\n    u_-(y) = a_- \\exp \\(\\frac{i}{2}\\<\\Gamma_-y,y\\>\\),\n  \\end{equation*}\nwhere $a_-\\in \\C$ and $\\Gamma_-$ is a complex symmetric $d\\times d$\nmatrix whose \nimaginary part is positive and non-degenerate. Consider $\\psi^\\eps$\nsolution to \\eqref{eq:lin}, with \\eqref{eq:asym-state}. Then the\nfollowing asymptotic expansion holds in $L^2(\\R^d)$:\n\\begin{equation*}\n  S^\\eps_{\\rm lin} \\psi_-^\\eps = \\frac{1}{\\eps^{d\/4}}e^{i\\delta^+\/\\eps} e^{ip^+\\cdot\n    (x-q^+)\/\\eps+iq^+\\cdot p^+\/(2\\eps)} \\hat R(G_+)\n  u_-\\(\\frac{x-q^+}{\\sqrt\\eps}\\) +\\O(\\sqrt\\eps),\n\\end{equation*}\nwhere $\\hat R(G_+)$ is the metaplectic transformation associated to\n$G_+ = \\frac{\\d (q^+,p^+)}{\\d(q^-,p^-)}$. \n\\end{theorem}\nAs a corollary, our main result yields another interpretation of the above\nstatement. It turns out that a complete scattering theory is available\nfor \\eqref{eq:ulin}. As a particular case of\nTheorem~\\ref{theo:scatt-class} (which addresses the nonlinear case), given $u_-\\in\n\\Sigma$, there exist a unique $u\\in C(\\R;\\Sigma)$ solution to\n\\eqref{eq:ulin} and a unique $u_+\\in\n\\Sigma$ such that  \n\\begin{equation*}\n  \\|e^{-i\\frac{t}{2}\\Delta}u(t)-u_\\pm \\|_\\Sigma \\Tend t {\\pm \\infty}\n  0. \n\\end{equation*}\nThen in the above theorem (where $u_-$ is restricted to be a Gaussian), we have\n\\begin{equation*}\n  u_+ = \\hat R(G_+)\n  u_-.\n\\end{equation*}\nFinally, we mention in passing the paper \\cite{NierENS}, where similar\nissues and results are obtained for\n\\begin{equation*}\n  i\\eps\\d_t \\psi^\\eps + \\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V\\(\\frac{x}{\\eps}\\) \\psi^\\eps + U(x)\\psi^\\eps,\n\\end{equation*}\nfor $V$ a short-range potential, and $U$ is bounded as well as its\nderivatives. The special scaling in $V$ implies that initially\nconcentrated waves (at scaled $\\eps$) first undergo the effects of $V$,\nthen exit a time layer of order $\\eps$, through which the main action of $V$\ncorresponds to the above quantum scattering operator (but with $\\eps=1$\ndue to the new scaling in the equation). Then, the action of $V$\nbecomes negligible, and the propagation of the wave is dictated by\nthe classical dynamics associated to $U$. \n\n\n\n\n\n\\subsection{Main results}\n\\label{sec:main}\n We now consider the nonlinear equation\n\\begin{equation}\n  \\label{eq:psi-eps}\n  i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps = V(x)\\psi^\\eps +\n  \\eps^\\alpha|\\psi^\\eps|^{2\\si}\\psi^\\eps,\n\\end{equation}\nalong with asymptotic data \\eqref{eq:asym-state}. We first prove that\nfor fixed $\\eps>0$, a scattering theory is available for\n\\eqref{eq:psi-eps}: at this stage, the value of $\\alpha$ is naturally\nirrelevant, as well as the form \\eqref{eq:asym-state}.\nTo establish a large data scattering theory for \\eqref{eq:psi}, we\nassume that the attractive part of the potential,\n\\begin{equation*}\n  (\\d_r V(x))_+=\n  \\(\\frac{x}{|x|}\\cdot \\nabla V(x)\\)_+\n\\end{equation*}\n is not too large, where $f_+=\\max (0,f)$ for any real number $f$.\n\\begin{theorem}\\label{theo:scatt-quant}\n  Let $d\\ge 3$, $\\frac{2}{d}<\\si<\\frac{2}{d-2}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. There exists $M=M(\\mu,d)$ such\n  that if the attractive part of the potential $(\\d_r V)_+$ satisfies\n  \\begin{equation*}\n    (\\d_r V(x))_+\\le \\frac{M}{(1+|x|)^{\\mu+1}},\\quad \\forall x\\in\n    \\R^d,\n  \\end{equation*}\none can define a\n  scattering operator for \\eqref{eq:psi} in $H^1(\\R^d)$: for\n  all $\\psi_-^\\eps\\in H^1(\\R^d)$, there exist a unique $\\psi^\\eps\\in\n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi} and a unique $\\psi_+^\\eps\\in\n  H^1(\\R^d)$ such that\n  \\begin{equation*}\n    \\|\\psi^\\eps(t)-e^{i\\frac{\\eps t}{2}\\Delta}\\psi_\\pm^\\eps\\|_{H^1(\\R^d)}\\Tend t {\\pm\n      \\infty} 0.\n  \\end{equation*}\nThe (quantum)  scattering operator is the map\n$S^\\eps:\\psi_-^\\eps\\mapsto \\psi_+^\\eps$.\n\\end{theorem}\nWe emphasize the fact that several recent results address the same\nissue, under various assumptions on the external potential $V$:\n\\cite{ZhZh14} treats the case where $V$ is an inverse square (a\nframework which is ruled out in our contribution), while in\n\\cite{CaDa-p}, the potential is more general than merely inverse\nsquare. In \\cite{CaDa-p}, a magnetic field is also included, and the\nLaplacian is perturbed with variable coefficients. We make more\ncomparisons with \\cite{CaDa-p} in Section~\\ref{sec:quant}. \n\\smallbreak\n\nThe second result of this paper concerns the scattering theory for the\nenvelope equation:\n\n\\begin{theorem}\\label{theo:scatt-class}\n   Let $d\\ge 1$, $\\frac{2}{d}\\le \\si<\\frac{2}{(d-2)_+}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>1$. One can define a\n  scattering operator for \\eqref{eq:u} in $\\Sigma$: for\n  all $u_-\\in \\Sigma$, there exist a unique $u\\in\n  C(\\R;\\Sigma)$ solution to \\eqref{eq:u} and a unique $u_+\\in\n  \\Sigma$ such that\n  \\begin{equation*}\n    \\|e^{-i\\frac{t}{2}\\Delta}u(t)-u_\\pm\\|_{\\Sigma}\\Tend t {\\pm\n      \\infty} 0.\n  \\end{equation*}\n\\end{theorem}\nAs mentioned above, the proof includes the construction of a linear\nscattering operator, comparing the dynamics associated to\n\\eqref{eq:ulin} to the free dynamics $e^{i\\frac{t}{2}\\Delta}$. In the\nabove formula, we have incorporated the information that\n$e^{i\\frac{t}{2}\\Delta}$ is unitary on $H^1(\\R^d)$, but \\emph{not on\n}$\\Sigma$ (see e.g. \\cite{CazCourant}). \n\\smallbreak\n\nWe can now state the nonlinear analogue to\nTheorem~\\ref{theo:version-lineaire}. Since\nTheorem~\\ref{theo:scatt-quant} requires $d\\ge 3$, we naturally have to\nmake this assumption. On the other hand, we will need the\napproximate envelope $u$ to be rather smooth, which requires a smooth\nnonlinearity, $\\si\\in \\N$. Intersecting this property with the\nassumptions of Theorem~\\ref{theo:scatt-quant} leaves only one case:\n$d=3$ and $\\si=1$, that is \\eqref{eq:1}, up to the scaling. We will\nsee in Section~\\ref{sec:cv} that considering $d=3$ is also crucial,\nsince the argument uses dispersive estimates which are known only in\nthe three-dimensional case for $V$ satisfying Assumption~\\ref{hyp:V}\nwith $\\mu>2$ (larger values for $\\mu$ could be considered in higher\ndimensions, though).  Introduce\nthe notation\n\\begin{equation*}\n  \\Sigma^k=\\{ f\\in H^k(\\R^d),\\quad x\\mapsto |x|^k f(x)\\in\n  L^2(\\R^d)\\}. \n\\end{equation*}\n\n\\begin{theorem}\\label{theo:cv}\n Let Assumptions~\\ref{hyp:V} and \\ref{hyp:flot} be satisfied, with\n $\\mu>2$ and $V$ as in Theorem~\\ref{theo:scatt-quant}. Consider \n $\\psi^\\eps$ solution to \n \\begin{equation*}\n    i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + \\eps^{5\/2}|\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n \\end{equation*}\nand such that \\eqref{eq:asym-state} holds, with $u_-\\in \\Sigma^7$.\nThen the\nfollowing asymptotic expansion holds in $L^2(\\R^3)$:\n\\begin{equation}\\label{eq:asym-finale}\n  S^\\eps \\psi_-^\\eps = \\frac{1}{\\eps^{3\/4}}e^{i\\delta^+\/\\eps} e^{ip^+\\cdot\n    (x-q^+)\/\\eps+iq^+\\cdot p^+\/(2\\eps)} u_+\\(\\frac{x-q^+}{\\sqrt\\eps}\\)\n  +\\O(\\sqrt\\eps), \n\\end{equation}\nwhere $S^\\eps$ is given by Theorem~\\ref{theo:scatt-quant} and $u_+$\nstems from Theorem~\\ref{theo:scatt-class}. \n\\end{theorem}\n\\begin{remark}\n  In the subcritical case, that is if we consider\n \\begin{equation*}\n    i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + \\eps^{\\alpha}|\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n \\end{equation*}\nalong with \\eqref{eq:asym-state}, for some $\\alpha>5\/2$, the argument\nof the proof shows that \\eqref{eq:asym-finale} remains true, but with $u_+$ given\nby the scattering operator associated to \\eqref{eq:ulin} (as opposed\nto \\eqref{eq:u}), that is, the same conclusion as in\nTheorem~\\ref{theo:version-lineaire} when $u_-$ is a Gaussian. \n\\end{remark}\nAs a corollary of the proof of the above result, and of the analysis\nfrom \\cite{CaFe11}, we infer:\n\\begin{corollary}[Asymptotic decoupling]\\label{cor:decoupling}\n  Let Assumption~\\ref{hyp:V} be satisfied, with\n $\\mu>2$ and $V$ as in Theorem~\\ref{theo:scatt-quant}. Consider \n $\\psi^\\eps$ solution to \n \\begin{equation*}\n    i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + \\eps^{5\/2}|\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n \\end{equation*}\nwith initial datum\n\\begin{equation*}\n  \\psi^\\eps(0,x) = \\sum_{j=1}^N\\frac{1}{\\eps^{3\/4}}a_j\\(\\frac{x-q_{0j}}{\\sqrt\\eps}\\)\n  e^{ip_{0j}\\cdot (x-q_{0j})\/\\eps}=:\\psi_0^\\eps(x),\n\\end{equation*}\nwhere $N\\ge 2$, $q_{0j},p_{0j}\\in \\R^3$, $p_{0j}\\not =0$ so that scattering   is\navailable as\n$t\\to +\\infty$ for  $(q_j(t),p_j(t))$, in the sense of\nAssumption~\\ref{hyp:flot}, and $a_j\\in \\Sch(\\R^3)$. We suppose\n$(q_{0j},p_{0j})\\not =(q_{0k},p_{0k})$ for $j\\not =k$. Then we have\nthe uniform estimate: \n\\begin{equation*}\n  \\sup_{t\\in \\R}\\left\\| \\psi^\\eps(t) - \\sum_{j=1}^N\n    \\varphi_j^\\eps(t)\\right\\|_{L^2(\\R^3)} \\Tend \\eps\n  0 0 ,\n\\end{equation*}\nwhere $\\varphi_j^\\eps$ is the approximate solution with the $j$-th\nwave packet as an initial datum. As a consequence, the \n asymptotic expansion holds in $L^2(\\R^3)$, as $\\eps \\to 0$:\n\\begin{equation*}\n  \\(W^\\eps_\\pm\\)^{-1} \\psi_0^\\eps =\\sum_{j=1}^N\n  \\frac{1}{\\eps^{3\/4}}e^{i\\delta_{j}^\\pm\/\\eps} e^{ip_{j}^\\pm\\cdot \n    (x-q_{j}^\\pm)\/\\eps+iq_{j}^\\pm\\cdot p_{j}^\\pm\/(2\\eps)}\n  u_{j\\pm}\\(\\frac{x-q_{j}^\\pm}{\\sqrt\\eps}\\) \n  +o(1), \n\\end{equation*}\nwhere the inverse wave operators $\\(W^\\eps_\\pm\\)^{-1} $ stem from\nTheorem~\\ref{theo:scatt-quant}, the $u_{j\\pm}$'s \nare the asymptotic states emanating from $a_j$, and \n\\begin{equation*}\n  \\delta_{j}^\\pm = \\lim_{t\\to \\pm\\infty}\\(S_j(t) - \\frac{q_j(t)\\cdot\n    p_j(t)-q_{0j}\\cdot p_{0j}}{2}\\)\\in \\R. \n\\end{equation*}\n\\end{corollary}\n\\begin{remark}\n  In the case $V=0$, the approximation by wave packets is actually\n  exact, since then $Q(t)\\equiv 0$, hence $u^\\eps=u$. For one wave\n  packet, Theorem~\\ref{theo:cv} \n  becomes empty, since it is merely a rescaling. On the other hand,\n  for two initial wave packets, even in the case $V=0$,\n  Corollary~\\ref{cor:decoupling} brings some information, reminiscent\n  of profile decomposition. More precisely, define $u^\\eps$ by\n  \\eqref{eq:chginc}, and choose (arbitrarily) to privilege the trajectory\n  $(q_1,p_1)$. The Cauchy problem is then equivalent to \n  \\begin{equation*}\n\\left\\{\n\\begin{aligned}\n    &i\\d_t u^\\eps+\\frac{1}{2}\\Delta u^\\eps = |u^\\eps|^2 u^\\eps,\\\\\n&    u^\\eps(0,y) = a_1(y) + a_2\\( y +\\frac{q_{01}-q_{02}}{\\sqrt\\eps}\\)\n    e^{ip_{02}\\cdot \\delta q_0\/\\eps -i\\delta p_0\\cdot y\/\\sqrt\\eps},\n  \\end{aligned}\n\\right.\n\\end{equation*}\nwhere we have set $\\delta p_0 = p_{01}-p_{02}$ and $\\delta q_0\n=q_{01}-q_{02}$. \nNote however that the initial datum is uniformly bounded in\n$L^2(\\R^3)$, but in no $H^s(\\R^3)$ for $s>0$ (if $p_{01}\\not =\np_{02}$), while the equation is \n$\\dot H^{1\/2}$-critical, Therefore, even in the case\n$V=0$, \nCorollary~\\ref{cor:decoupling} does not seem to be a consequence of\nprofile decompositions like in\ne.g. \\cite{DuHoRo08,Keraani01,MerleVega98}. In view of\n\\eqref{eq:hamil}, the approximation provided by\nCorollary~\\ref{cor:decoupling} reads, in that case:\n\\begin{equation*}\n  u^\\eps(t,y) = u_1(t,y) + u_2\\(t,y\n  +\\frac{t \\delta p_0+\\delta q_0}{\\sqrt\\eps}\\)\n  e^{i\\phi_2^\\eps(t,y)}+o(1)\\quad \\text{in }L^\\infty(\\R;L^2(\\R^3)),\n\\end{equation*}\nwhere the phase shift is given by\n\\begin{align*}\n  \\phi^\\eps_2(t,y) &= \\frac{1}{\\eps}p_{02}\\cdot \\( t\\delta p_0+\\delta\n  q_0\\) -\\frac{1}{\\sqrt\\eps}\\delta p_0\\cdot y +\\frac{t}{2\\eps}\n  \\( |p_{02}|^2-|p_{01}|^2\\) \\\\\n&= \\frac{1}{\\eps}p_{02}\\cdot \\delta\n  q_0 -\\frac{1}{\\sqrt\\eps}\\delta p_0\\cdot y -\\frac{t}{2\\eps}|\\delta\n  p_0|^2. \n\\end{align*}\n\\end{remark}\n\n\n\n\\noindent {\\bf Notation.} We write $a^\\eps(t)\\lesssim b^\\eps(t)$\nwhenever there exists $C$ independent of $\\eps\\in (0,1]$ and $t$ such\nthat $a^\\eps(t)\\le C b^\\eps(t)$. \n \n\n\n\n\n\n\n\n\n\n\\section{Spectral properties and consequences}\n\\label{sec:spectral}\n\nIn this section, we derive some useful properties for the Hamiltonian\n\\begin{equation*}\n  H=-\\frac{1}{2}\\Delta +V.\n\\end{equation*}\nSince the dependence upon $\\eps$ is not addressed in this\nsection, we assume $\\eps=1$.\n\\smallbreak\n\nFirst, it follows for instance from \\cite{Mourre} that\nAssumption~\\ref{hyp:V} implies that $H$ has no\nsingular spectrum. Based on Morawetz estimates, we show that $H$ has\nno eigenvalue, provided that the  attractive part of $V$ is\nsufficiently small. Therefore, the spectrum of $H$ is\npurely absolutely continuous. \nFinally, again if  the  attractive part of $V$ is\nsufficiently small, zero is not a resonance of $H$, so Strichartz\nestimates are available for $e^{-itH}$. \n\n\n\n\n\\subsection{Morawetz estimates and a first consequence}\n\\label{sec:morawetz}\n\nIn this section, we want to treat both linear and nonlinear equations,\nso we consider\n\\begin{equation}\n  \\label{eq:psi-gen}\n  i\\d_t \\psi +\\frac{1}{2}\\Delta \\psi = V\\psi + \\lambda\n  |\\psi|^{2\\si}\\psi,\\quad \\l \\in \\R.\n\\end{equation}\nMorawetz estimate in the linear case $\\l=0$ will show the absence of\neigenvalues. In the nonlinear case $\\l>0$, these estimates will be a\ncrucial tool for prove scattering in the quantum case. \n The following lemma and its proof are essentially a rewriting of the\n presentation from \\cite{BaRuVe06}.  \n\\begin{proposition}[Morawetz inequality]\\label{prop:Morawetz}\n   Let $d\\ge 3$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. There exists $M=M(\\mu,d)>0$ such\n  that if the attractive part of the potential satisfies\n  \\begin{equation*}\n    (\\d_r V(x))_+ \\le \\frac{M}{(1+|x|)^{\\mu+1}},\\quad \\forall x\\in\n    \\R^d,\n  \\end{equation*}\nthen any solution $\\psi\\in L^\\infty(\\R;H^1(\\R^d))$ to \\eqref{eq:psi-gen}\nsatisfies\n\\begin{equation}\\label{eq:morawetz}\n \\l  \\iint_{\\R\\times \\R^d}\\frac{|\\psi(t,x)|^{2\\si+2}}{|x|}dtdx +\n \\iint_{\\R\\times \\R^d}\\frac{|\\psi(t,x)|^{2}}{(1+|x|)^{\\mu+1}}dtdx\\lesssim \n  \\|\\psi\\|_{L^\\infty(\\R;H^1)}^2. \n\\end{equation}\n\\end{proposition}\nIn other words, the main obstruction to global dispersion for $V$\ncomes from $(\\d_r V)_+$, which is the attractive contribution of $V$\nin classical trajectories, while $(\\d_r V)_-$ is the repulsive part,\nwhich does not ruin the dispersion associated to $-\\Delta$ (it may\n reinforce it, see e.g. \\cite{CaDCDS}, but repulsive\npotentials do not necessarily improve the dispersion, see \\cite{GoVeVi06}). \n\\begin{proof}\n  The proof follows standard arguments, based on virial identities\n  with a suitable weight. We resume the main steps of the\n  computations, and give more details on the choice of the weight in\n  our context. For a real-valued function $h(x)$, we compute, for $\\psi$ solution\n  to \\eqref{eq:psi},\n  \\begin{equation*}\n    \\frac{d}{dt}\\int h(x)|\\psi(t,x)|^2dx = \\IM \\int \\bar \\psi(t,x) \\nabla\n    h(x)\\cdot \\nabla \\psi(t,x)dx,\n  \\end{equation*}\n  \\begin{equation}\n    \\label{eq:viriel}\n    \\begin{aligned}\n        \\frac{d}{dt}\\IM \\int \\bar \\psi(t,x) \\nabla\n    h(x)\\cdot \\nabla \\psi(t,x)dx &= \\int \\nabla \\bar \\psi(t,x)\\cdot\n    \\nabla^2h(x)\\nabla \\psi(t,x)dx \\\\\n-\\frac{1}{4}\\int |\\psi(t,x)|^2 &\\Delta^2\n    h(x)dx -\\int |\\psi(t,x)|^2\\nabla V\\cdot \\nabla h(x)dx\\\\\n &   +\\frac{\\l\\si}{\\si+1}\\int |\\psi(t,x)|^{2\\si+2}\\Delta h(x) dx. \n \\end{aligned}\n   \\end{equation}\nIn the case $V=0$, the standard choice is $h(x)=|x|$, for which\n\\begin{equation*}\n  \\nabla h=\\frac{x}{|x|},\\quad \\nabla^2_{jk}h =\n  \\frac{1}{|x|}\\(\\delta_{jk}-\\frac{x_jx_k}{|x|^2}\\),\\quad \\Delta h\\ge\n  \\frac{d-1}{h},\\quad \\text{and }\\Delta^2 h\\le 0\\text{ for }d\\ge 3. \n\\end{equation*}\nThis readily yields Proposition~\\ref{prop:Morawetz} in the repulsive case\n$\\d_r V\\le 0$, since $\\nabla h\\in L^\\infty$. \n\\smallbreak\n\nIn the same spirit as in \\cite{BaRuVe06}, we proceed by perturbation to\nconstruct a suitable weight when the attractive part of the potential\nis not too large. We seek a priori a radial weight, $h=h(|x|)\\ge 0$, so we\nhave \n\\begin{align*}\n & \\Delta h = h'' +\\frac{d-1}{r} h',\\\\\n& \\Delta^2 h =  h^{(4)}\n  +2\\frac{d-1}{r} h^{(3)} +\\frac{(d-1)(d-3)}{r^2}h'' -\n  \\frac{(d-1)(d-3)}{r^3}h',\\\\\n&\\nabla^2_{jk} h = \\frac{1}{r}\\(\\delta_{jk}-\\frac{x_jx_k}{r^2}\\) h'\n+\\frac{x_jx_k}{r^2}h''. \n\\end{align*}\nWe construct a function $h$ such that $h',h''\\ge 0$, so the condition\n$\\nabla^2 h\\ge 0$ will remain. The goal is then to construct a radial\nfunction $h$ such that the second line in \\eqref{eq:viriel} is\nnon-negative, along with $\\Delta h \\ge \\eta\/|x|$ for some $\\eta>0$.\n\\smallbreak\n\n\\noindent {\\bf Case $d=3$.} In this case, the expression for $\\Delta^2\nh$ is simpler, and the above conditions read\n\\begin{align*}\n &\\frac{1}{4} h^{(4)}\n  +\\frac{1}{r} h^{(3)} + \\nabla V(x)\\cdot \\nabla h\\le 0,\\\\\n& h''+\\frac{2}{r}h'\\ge \\frac{\\eta}{r},\\quad h',h''\\ge 0. \n\\end{align*}\nSince we do not suppose a priori that $V$ is a radial potential, the\nfirst condition is not rigorous. We actually use the fact that for\n$h'\\ge 0$, Assumption~\\ref{hyp:V} implies\n\\begin{equation*}\n  \\nabla V(x)\\cdot \\nabla h \\le \\(\\d_r V(x)\\)_+ h'(r)\\le\n  \\frac{M}{(1+r)^{\\mu+1}}h'(r). \n\\end{equation*}\nTo achieve our goal, it is therefore sufficient to require:\n\\begin{align}\n \\label{eq:h1}&\\frac{1}{4} h^{(4)}\n  +\\frac{1}{r} h^{(3)} +  \\frac{M}{(1+r)^{\\mu+1}}h' \\le 0,\\\\\n\\label{eq:h2}& h''+\\frac{2}{r}h'\\ge \\frac{\\eta}{r},\\quad h'\\in L^\\infty(\\R_+), \\\nh',h''\\ge 0. \n\\end{align}\nIn view of \\eqref{eq:h2}, we seek\n\\begin{equation*}\n  h'(r) = \\eta +\\int_0^r h''(\\rho)d\\rho.\n\\end{equation*}\nTherefore, if $h''\\ge 0$ with $h''\\in L^1(\\R_+)$, \\eqref{eq:h2} will\nbe automatically fulfilled. We now turn to \\eqref{eq:h1}. Since we\nwant $h'\\in L^\\infty$, we may even replace $h'$ by a constant in\n\\eqref{eq:h1}, and solve, for $C>0$, the ODE\n\\begin{equation*}\n  \\frac{1}{4} h^{(4)}\n  +\\frac{1}{r} h^{(3)} +  \\frac{C}{(1+r)^{\\mu+1}}=0.\n\\end{equation*}\nWe readily have\n\\begin{equation*}\n  h^{(3)}(r) = -\\frac{4C}{r^4}\\int_0^r\\frac{\\rho^4}{(1+\\rho)^{\\mu+1}}d\\rho,\n\\end{equation*}\nalong with the properties $h^{(3)}(0)=0$, \n\\begin{equation*}\n  h^{(3)}(r)\\Eq r \\infty -\\frac{k}{r^{\\min (\\mu,4)}},\\quad \\text{for some }k>0.\n\\end{equation*}\nIt is now natural to set\n\\begin{equation*}\n  h''(r) = -\\int_r^\\infty h^{(3)}(\\rho)d\\rho,\n\\end{equation*}\nso we have $h''\\in C([0,\\infty);\\R_+)$ and\n\\begin{equation*}\n  h''(r) \\Eq r \\infty \\frac{\\kappa}{r^{\\min (\\mu-1,3)}},\\quad \\text{for some }\\kappa>0.\n\\end{equation*}\nThis function is indeed in $L^1$ if and only if $\\mu>2$. We \ndefine $h$ by  $h(r)= \\int_0^r h'(\\rho)d\\rho$,\n\\begin{equation}\\label{eq:h3}\n   h^{(3)}(r) = -\\frac{K}{r^4}\\int_0^r\\frac{\\rho^4}{(1+\\rho)^{\\mu+1}}d\\rho,\n\\end{equation}\nfor some $K>0$, $h''$ and $h'$ being given by the above relations:\n\\eqref{eq:h2} is satisfied for any value of $K>0$, and \\eqref{eq:h1}\nboils down to an inequality of the form\n\\begin{equation}\\label{eq:M}\n  -\\frac{K}{4} +M\\(\\eta +C(\\mu)K\\)\\le 0,\n\\end{equation}\nwhere $C(\\mu)$ is proportional to \n\\begin{equation*}\n \\frac{1}{K} \\|h'\\|_{L^\\infty} = \\int_0^\\infty \\int_r^\\infty\n \\frac{1}{\\rho^4}\\int_0^\\rho \\frac{s^4}{(1+s)^{\\mu+1}}dsd\\rho dr.\n\\end{equation*}\nWe infer that \\eqref{eq:h3} is satisfied for $K\\gg \\eta$, provided\nthat $M<\\frac{1}{4C(\\mu)}$. Note then that by construction, we may also\nrequire\n\\begin{equation*}\n  \\frac{1}{4}\\Delta^2 h +\\nabla V\\cdot \\nabla h\\le \\frac{-c_0}{(1+|x|)^{\\mu+1}},\n\\end{equation*}\nfor $c_0>0$ morally very small. \n\\smallbreak\n\n\\noindent {\\bf Case $d\\ge 4$.} Resume the above reductions, pretending\nthat the last two terms in $\\Delta^2 h$ are not present: \\eqref{eq:h3}\njust becomes\n\\begin{equation*}\n   h^{(3)}(r) = -\\frac{K}{r^{2d-2}}\\int_0^r\\frac{\\rho^{2d-2}}{(1+\\rho)^{\\mu+1}}d\\rho,\n\\end{equation*}\nand we see that with $h''$ and $h'$ defined like before, we have\n\\begin{equation*}\n  rh''-h'= -\\eta- \\int_0^r h'' +rh''.\n\\end{equation*}\nSince this term is negative at $r=0$ and has a non-positive\nderivative, we have $rh''-h'\\le 0$, so finally $\\Delta^2 h\\le 0$. \n\\end{proof}\n\nWe infer that $H$ has no eigenvalue. Indeed, if there were an $L^2$ solution\n$\\psi=\\psi(x)$ \nto $H\\psi =E\\psi$, $E\\in \\R$, then $\\psi\\in\nH^2(\\R^d)$, and $\\psi(x)e^{-iEt}$ would be an $H^1$\nsolution to \\eqref{eq:psi-gen} for $\\l=0$. This is contradiction with\nthe global integrability in time from \\eqref{eq:morawetz}, so\n$\\si_{\\rm pp}(H)=\\emptyset$. \n\n\\subsection{Strichartz estimates}\n\\label{sec:strichartz}\n\nIn\n\\cite[Proposition~3.1]{BaRuVe06}, it is proved that zero is\nnot a resonance of $H$, but with a definition of resonance which is\nnot quite the definition in \\cite{RodnianskiSchlag}, which contains a\nresult that we want to use. So we shall resume the argument.\n\nBy definition (as in \\cite{RodnianskiSchlag}), zero is a resonance of\n$H$, if there is a distributional solution \n$\\psi\\not\\in L^2$, such that  $\\<x\\>^{-s}\\psi\\in L^2(\\R^d)$ for all\n$s>\\frac{1}{2}$, to\n$H\\psi=0$.  \n\\begin{corollary}\n  Under the assumptions of Proposition~\\ref{prop:Morawetz}, zero is not a\n  resonance of $H$.\n\\end{corollary}\n\\begin{proof}\n  Suppose that zero is a resonance of $H$. Then by definition, we\n  obtain a stationary  distributional solution of \\eqref{eq:psi-gen} (case\n  $\\l=0$), $\\psi= \\psi(x)$, and we may assume that it is\n  real-valued. Since $\\Delta \\psi = \n  2V\\psi$, Assumption~\\ref{hyp:V} implies\n  \\begin{equation*}\n    \\<x\\>^{\\mu-s}\\Delta \\psi\\in L^2(\\R^d),\\quad \\forall s>\\frac{1}{2}. \n  \\end{equation*}\nThis implies that $\\nabla \\psi\\in L^2$, by taking for instance $s=1$ in\n\\begin{equation*}\n  \\int|\\nabla \\psi|^2 = -\\int \\<x\\>^{-s}\\psi \\<x\\>^s\\Delta \\psi.\n\\end{equation*}\nBy definition, for all test function $\\varphi$,\n\\begin{equation}\\label{eq:variat}\n  \\frac{1}{2}\\int_{\\R^d}\\nabla \\varphi(x) \\cdot \\nabla \\psi(x)dx\n  +\\int_{\\R^d}V(x)\\varphi(x)\\psi(x)dx =0.\n\\end{equation}\nLet $h$ be the weight constructed in the proof of\nProposition~\\ref{prop:Morawetz}, and consider\n\\begin{equation*}\n  \\varphi = \\psi\\Delta h +2\\nabla \\psi\\cdot \\nabla h. \n\\end{equation*}\nSince $\\nabla h\\in L^\\infty$, $\\nabla^2 h(x)=\\O(\\<x\\>^{-1})$, and\n$\\nabla^3 h(x)= \\O(\\<x\\>^{-2})$, we see\nthat $\\varphi \\in H^1$, and that this choice is allowed in\n\\eqref{eq:variat}. Integration by parts then yields \\eqref{eq:viriel}\n(where the left hand side is now zero):\n\\begin{equation*}\n  0=\\int \\nabla \\psi\\cdot \\nabla^2h \\nabla \\psi -\\frac{1}{4}\\int\n  \\psi^2\\Delta^2 h -\\int \\psi^2 \\nabla V\\cdot \\nabla h.\n\\end{equation*}\nBy construction of $h$, this implies\n\\begin{equation*}\n  \\int_{\\R^d}\\frac{\\psi(x)^2}{(1+|x|)^{\\mu+1}}dx\\le 0,\n\\end{equation*}\nhence $\\psi\\equiv 0$. \n\\end{proof}\nTherefore,\n\\cite[Theorem~1.4]{RodnianskiSchlag} implies non-endpoint global in\ntime Strichartz estimates.  In the case $d=3$, we know from\n\\cite{Go06} that (in view of the above spectral properties)\n\\begin{equation*}\n  \\|e^{-itH}\\|_{L^1\\to L^\\infty}\\le C |t|^{-d\/2},\\quad \\forall t\\not\n  =0,\n\\end{equation*}\na property which is stronger than Strichartz estimates, and yields the\nendpoint Strichartz estimate missing in \\cite{RodnianskiSchlag}, from\n\\cite{KT}. On the other \nhand, this dispersive estimate does not seem to be known under\nAssumption~\\ref{hyp:V} with $\\mu>2$ when $d\\ge 4$: stronger assumptions\nare always present so far (see e.g. \\cite{CaCuVo09,ErGr10}). However,\nendpoint Strichartz estimates for $d\\ge 4$ are a consequence of\n\\cite[Theorem~1.1]{AFVV10}, under the assumptions of\nProposition~\\ref{prop:Morawetz}. \n\n\\begin{proposition}\\label{prop:StrichartzRS}\nLet $d\\ge 3$. Under the assumptions of\nProposition~\\ref{prop:Morawetz}, for all $(q,r)$ such that \n  \\begin{equation}\\label{eq:adm}\n    \\frac{2}{q}=d\\(\\frac{1}{2}-\\frac{1}{r}\\),\\quad 2<q\\le \\infty,\n  \\end{equation}\nthere exists $C=C(q,d)$ such that \n\\begin{equation*}\n  \\|e^{-itH} f\\|_{L^q(\\R;L^r(\\R^d))}\\le C \\|f\\|_{L^2(\\R^d)},\\quad\n  \\forall f\\in L^2(\\R^d). \n\\end{equation*}\n\\end{proposition}\nIt is classical that this homogeneous Strichartz estimate, a duality\nargument and Christ-Kiselev's Theorem imply the inhomogeneous\ncounterpart. For two admissible pairs $(q_1,r_1)$ and $(q_2,r_2)$\n(that is, satisfying \\eqref{eq:adm}), there exists $C_{q_1,q_2}$\nindependent of the time interval $I$ such\nthat if we denote by\n\\begin{equation*}\n  R(F)(t,x) = \\int_{I\\cap \\{s\\le t\\}} e^{-i(t-s)H}F(s,x)ds,\n\\end{equation*}\nwe have\n\\begin{equation*}\n  \\|R(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^d))}\\le\n  C_{q_1,q_2}\\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^d))},\\quad \\forall F\\in\n  L^{q_2'}(I;L^{r_2'}(\\R^d)). \n\\end{equation*}\n\\smallbreak\n\nNote that the assumption $\\mu>2$ seems essentially sharp in order to have\nglobal in time Strichartz estimates. The result remains true for $\\mu\n=2$ (\\cite{BPST03,BPST04}), but in \\cite{GoVeVi06}, the authors\nprove that for repulsive potentials which are homogeneous of degree\nsmaller than $2$, global Strichartz estimates fail to exist.\n\n\n\n\n\n\\section{Quantum scattering}\n\\label{sec:quant}\n\nIn this section, we prove Theorem~\\ref{theo:scatt-quant}. Since the\ndependence upon $\\eps$ is not measured in\nTheorem~\\ref{theo:scatt-quant}, we shall \nconsider the case $\\eps=1$, corresponding to \n\\begin{equation}\n  \\label{eq:psi}\n  i\\d_t \\psi +\\frac{1}{2}\\Delta \\psi = V\\psi + |\\psi|^{2\\si}\\psi.\n\\end{equation}\nWe split the proof of Theorem~\\ref{theo:scatt-quant} into two\nsteps. First, we solve the Cauchy problem with data prescribed at\n$t=-\\infty$, that is, we show the existence of wave operators. Then,\ngiven an initial datum at $t=0$, we show that the (global) solution to\n\\eqref{eq:psi} behaves asymptotically like a free solution, which\ncorresponds to asymptotic completeness. \n\\smallbreak\n\nFor each of these two steps, we first show that the nonlinearity is\nnegligible for large time, and then recall that the potential is\nnegligible for large time (linear scattering). This means that for any $\\tilde \\psi_-\\in\nH^1(\\R^d)$, there exists  a unique $\\psi\\in \n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi}  such that\n  \\begin{equation*}\n    \\|\\psi(t)-e^{-itH}\\tilde \\psi_-\\|_{H^1(\\R^d)}\\Tend t {-\n      \\infty} 0,\n  \\end{equation*}\nand for any $\\varphi\\in H^1(\\R^d)$, there exist a unique  $\\psi\\in\n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi} and a unique $\\tilde\\psi_+\\in\n  H^1(\\R^d)$ such that\n\\begin{equation*}\n    \\|\\psi(t)-e^{-itH}\\tilde \\psi_+\\|_{H^1(\\R^d)}\\Tend t {+\n      \\infty} 0.\n  \\end{equation*}\nThen, we recall that the potential $V$ is negligible for large\ntime. We will adopt the following notations for the propagators,\n\\begin{equation*}\n  U(t)=e^{i\\frac{t}{2}\\Delta},\\quad U_V(t)= e^{-itH}. \n\\end{equation*}\n\n\n\nIn order to construct wave operators which show that the nonlinearity\ncan be neglected for large time, we shall work with an $H^1$\nregularity, on the Duhamel's formula associated to \\eqref{eq:psi} in\nterms of $U_V$, with a prescribed asymptotic behavior as $t\\to\n-\\infty$:\n\\begin{equation}\n  \\label{eq:duhamel-}\n  \\psi(t) = U_V(t)\\tilde \\psi_- -i\\int_{-\\infty}^t\n  U_V(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds. \n\\end{equation}\nApplying the gradient to this formulation brings up the problem of\nnon-commutativity with $U_V$. The worst term is actually the linear\none, $U_V(t)\\tilde \\psi_-$, since\n\\begin{equation*}\n  \\nabla \\(U_V(t)\\tilde \\psi_-\\) = U_V(t)\\nabla \\tilde \\psi_-\n  -i\\int_0^t U_V(t-s)\\((U_V(s)\\tilde \\psi_-)\\nabla V\\)ds.\n\\end{equation*}\nSince the construction of wave operators relies on the use of\nStrichartz estimates, it would be necessary to have an estimate of\n\\begin{equation*}\n \\left\\|\\nabla \\(U_V(t)\\tilde \\psi_-\\)\\right\\|_{L^qL^r}\n\\end{equation*}\nin terms of $\\psi_-$, for admissible pairs\n$(q,r)$. Proposition~\\ref{prop:StrichartzRS} yields\n\\begin{equation*}\n  \\left\\|\\nabla \\(U_V(t)\\tilde \\psi_-\\)\\right\\|_{L^qL^r} \\lesssim \\|\\nabla \\tilde\n  \\psi_-\\|_{L^2} + \\|(U_V(t)\\tilde \\psi_-)\\nabla V\\|_{L^{\\tilde\n      q'}L^{\\tilde r'}},\n\\end{equation*}\nfor any admissible pair $(\\tilde q,\\tilde r)$. In the last factor,\ntime is present only in the term $U_V(t)\\tilde \\psi_-$, so to be able\nto use Strichartz estimates again, we need to consider $\\tilde\nq=2$, in which case $\\tilde r=2^*:=\\frac{2d}{d-2}$:\n\\begin{equation*}\n  \\|(U_V(t)\\tilde \\psi_-)\\nabla V\\|_{L^2L^{{2^*}'}}\\le \\|U_V(t)\\tilde\n  \\psi_-\\|_{L^2L^{2^*}}\\|\\nabla V\\|_{L^{d\/2}},\n\\end{equation*}\nwhere Assumption~\\ref{hyp:V} implies $\\nabla V\\in L^{d\/2}(\\R^d)$ as\nsoon as $\\mu>1$. Using the endpoint Strichartz estimate from\nProposition~\\ref{prop:StrichartzRS}, we have\n\\begin{equation*}\n  \\|U_V(t)\\tilde\n  \\psi_-\\|_{L^2L^{2^*}} \\lesssim \\|\\tilde \\psi_-\\|_{L^2}, \n\\end{equation*}\nand we have:\n\\begin{lemma}\\label{lem:stri2}\n  Let $d\\ge 3$. Under the assumptions of\n  Proposition~\\ref{prop:Morawetz}, for all admissible pair $(q,r)$, \n  \\begin{equation*}\n    \\|e^{-itH}f\\|_{L^q(\\R;W^{1,r}(\\R^d))}\\lesssim \\|f\\|_{H^1(\\R^d)}. \n  \\end{equation*}\n\\end{lemma}\nWe shall rather use a vector-field, for we believe this approach may be\ninteresting in other contexts.\n\n\n\\subsection{Vector-field}\n\\label{sec:vector-field}\n\n We\nintroduce a vector-field which naturally commutes with $U_V$, and\nis comparable with the gradient. \n\\smallbreak\n\nFrom Assumption~\\ref{hyp:V}, $V$ is bounded, so there exists $c_0\\ge\n0$ such that $V+c_0\\ge 0$. We shall consider the operator\n\\begin{equation*}\n  A = \\sqrt{H+c_0}=\\sqrt{-\\frac{1}{2}\\Delta +V+c_0}.\n\\end{equation*}\n\\begin{lemma}\\label{lem:A}\n Let $d\\ge 3$, and $V$ satisfying Assumption~\\ref{hyp:V} with\n $V+c_0\\ge 0$. For every $1<r<\\infty$, there exists $C_r,K_r$  such\n that for all $f\\in    W^{1,r}(\\R^d)$,\n  \\begin{equation}\n    \\label{eq:A}\n    \\|A f\\|_{L^r}\\le C_r\n    \\(\\|f\\|_{L^r}+\\|\\nabla f\\|_{L^r}\\)\\le K_r \\(\\|f\\|_{L^r}+\\|A f\\|_{L^r}\\) .\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  The first inequality is very close to \\cite[Theorem~1.2]{AFVV10}, and the\n  proof can readily be adapted. On the other hand, the second\n  inequality would require the restriction $4\/3<r<4$ if we followed \n  the same approach, based on  Stein's interpolation theorem (a\n  similar approach for followed in e.g. \\cite{KiViZh09}). We\n  actually take  advantage of the smoothness of the potential $V$ to\n  rather apply Calder\\'on--Zygmund result on the action of\n  pseudo-differential operators. \n\\smallbreak\n\nWe readily check that the two functions\n\\begin{equation*}\n  a(x,\\xi)=\n  \\sqrt{\\frac{\\frac{|\\xi|^2}{2}+V(x)+c_0}{1+|\\xi|^2}},\\quad\n  b(x,\\xi) = \\sqrt{\\frac{|\\xi|^2}{\\frac{|\\xi|^2}{2}+V(x)+c_0+1}} ,\n\\end{equation*}\nare symbols of order zero, in the sense\nthat they satisfy\n\\begin{equation*}\n  |\\d_x^\\alpha\\d_\\xi^\\beta a(x,\\xi)| +  |\\d_x^\\alpha\\d_\\xi^\\beta\n  b(x,\\xi)| \\le C_{\\alpha,\\beta}\\<\\xi\\>^{-|\\beta|},\n\\end{equation*}\nfor all $\\alpha,\\beta\\in \\N^d$. This implies that the\npseudo-differential operators of symbol \n$a$ and $b$, respectively, are bounded on $L^r(\\R^d)$, for\nall $1<r<\\infty$; see e.g. \\cite[Theorem~5.2]{Taylor3}. In the case of\n$a$, this yields the first inequality in \\eqref{eq:A}, and in the case of $b$, this\nyields the second inequality. \n\\end{proof}\n\n\\subsection{Wave operators}\n\\label{sec:wave-op}\n\nWith the tools presented in the previous section, we can prove the\nfollowing result by adapting the standard proof of the case $V=0$, as\nestablished in \\cite{GV85}.\n\\begin{proposition}\\label{prop:waveop-quant}\n  Let $d\\ge 3$, $\\frac{2}{d}\\le \\si<\\frac{2}{d-2}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. For\n  all $\\tilde \\psi_-\\in H^1(\\R^d)$, there exists a unique \n$$\\psi\\in\n  C((-\\infty,0];H^1(\\R^d))\\cap\n  L^{\\frac{4\\si+4}{d\\si}}((-\\infty,0);L^{2\\si+2}(\\R^d))$$\n solution to    \\eqref{eq:psi}  such that \n  \\begin{equation*} \n    \\|\\psi(t)-e^{-it H}\\tilde\\psi_-\\|_{H^1(\\R^d)}\\Tend t {-\n      \\infty} 0.\n  \\end{equation*} \n\\end{proposition}\n\\begin{proof}\n  The main part of the proof is to prove that \\eqref{eq:duhamel-} has\n  a fixed point. Let\n  \\begin{equation*}\n   q=\\frac{4\\si+4}{d\\si}.\n  \\end{equation*}\nThe pair $(q,2\\si+2)$ is admissible, in the sense that it satisfies\n\\eqref{eq:adm}.  \nWith the notation $L^\\beta_TY=L^\\beta(]-\\infty,-T];Y)$, we\n  introduce:  \n  \\begin{align*}\n    X_T:=\\Big\\{ \\psi\\in C(]-\\infty,-T];H^1)\\ ;\\ &\\left\\|\n    \\psi\\right\\|_{L^q_TL^{2\\si+2}} \\le  K\\|\\tilde\n                                                  \\psi_-\\|_{L^2},\\\\\n\\left\\|\n    \\nabla \\psi\\right\\|_{L^q_TL^{,2\\si+2}} \\le  K\\|\\tilde\n                                                  \\psi_-\\|_{H^1},\\quad &\n \\left\\|  \\psi\\right\\|_{L^\\infty_TL^2} \\le 2 \\|\\tilde \\psi_-\\|_{L^2}\\,\n    ,\\\\\n \\left\\| \\nabla  \\psi\\right\\|_{L^\\infty_TL^2} \\le K \\|\\tilde \\psi_-\\|_{H^1}\n,\\quad \n&\\left\\|  \\psi\\right\\|_{L^q_T L^{2\\si+2}} \\le 2 \\left\\|\n    U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^q_T L^{2\\si+2}}\\Big\\},\n  \\end{align*}\nwhere $K$ will be chosen sufficiently large in terms of the\nconstants present in Strichartz estimates presented in\nProposition~\\ref{prop:StrichartzRS}. Set\n$r=s=2\\si +2$: we have\n\\begin{equation*}\n  \\frac{1}{r'}= \\frac{1}{r}+\\frac{2\\si}{s},\\quad\n\\frac{1}{q'}= \\frac{1}{q}+\\frac{2\\si}{k},\n\\end{equation*}\nwhere $q\\le k<\\infty$ since $2\/d\\le\n\\si<2\/(d-2)$. Denote by $\\Phi(\\psi)$ the right hand side of\n\\eqref{eq:duhamel-}. For $\\psi\\in X_T$, Strichartz estimates and H\\\"older\ninequality yield, for all admissible pairs $(q_1,r_1)$:\n\\begin{align*}\n  \\left\\| \\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} &\\le C_{q_1}\\|\\tilde    \n\\psi_-\\|_{L^2} +  C\\left\\| |\\psi|^{2\\si}\\psi\\right\\|_{L^{q'}_TL^{r'}}  \n  \\\\\n&\\le C_{q_1}\\|\\tilde \\psi_-\\|_{L^2}  +\nC\\|\\psi\\|_{L^k_TL^s}^{2\\si}\\|\\psi\\|_{L^q_T L^r}\\\\\n&\\le C_{q_1}\\|\\tilde \\psi_-\\|_{L^2} + C\\|\\psi\\|_{L^q_TL^r}^{2\\si\\theta\n  }\\|\\psi\\|_{L^\\infty_TL^r}^{2\\si(1-\\theta) } \\|\\psi\\|_{L^q_T L^r} ,\n\\end{align*}\nfor some $0<\\theta\\le 1$, where we have used the property\n$r=s=2\\si+2$. Sobolev embedding and the definition of $X_T$ then imply:\n\\begin{align*}\n \\left\\| \\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} \\le C_{q_1}\\|\\tilde\n  \\psi_-\\|_{L^2} + C\\left\\|U_V(\\cdot) \\tilde \n  \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta \n  }\\|\\psi\\|_{L^\\infty_TH^1}^{2\\si(1-\\theta) } \\|\\psi\\|_{L^q_T L^r} .\n\\end{align*}\nWe now apply the operator $A$. Since $A$ commutes with $H$, we have\n\\begin{equation*}\n  \\left\\| A\\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} \\lesssim\n  \\|A\\tilde \\psi_-\\|_{L^2} + \\left\\|\n    A\\(|\\psi|^{2\\si}\\psi\\)\\right\\|_{L^{q'}_TL^{r'}}. \n\\end{equation*}\nIn view of Lemma~\\ref{lem:A}, we have successively,\n\\begin{align*}\n  \\|A\\tilde \\psi_-\\|_{L^2} &\\lesssim \\|\\tilde \\psi_-\\|_{H^1},\\\\\n \\left\\|\n    A\\(|\\psi|^{2\\si}\\psi\\)\\right\\|_{L^{q'}_TL^{r'}}&\\lesssim   \\left\\|\n    |\\psi|^{2\\si}\\psi\\right\\|_{L^{q'}_TL^{r'}} + \\left\\|\n  \\nabla   \\(|\\psi|^{2\\si}\\psi\\)\\right\\|_{L^{q'}_TL^{r'}} \\\\\n&\\lesssim \\|\\psi\\|_{L^k_TL^s}^{2\\si}\\(\\|\\psi\\|_{L^q_T L^r} +\n  \\|\\nabla\\psi\\|_{L^q_T L^r} \\)\\\\\n&\\lesssim \\|\\psi\\|_{L^k_TL^s}^{2\\si}\\(\\|\\psi\\|_{L^q_T L^r} +\n  \\|A\\psi\\|_{L^q_T L^r} \\).\n\\end{align*}\nWe infer along the same lines as above,\n\\begin{align*}\n \\left\\| \\nabla\\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} &\\lesssim\n  \\|\\tilde \\psi_-\\|_{H^1} +\\left\\|U_V(\\cdot) \\tilde\n  \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta \n  }\\|\\psi\\|_{L^\\infty_TH^1}^{2\\si(1-\\theta) } \\( \n\\| \\psi\\|_{L^q_T L^r} + \\|A\\psi\\|_{L^q_T L^r}\\) .\n \\end{align*}\nWe have also\n\\begin{align*}\n\\left\\| \\Phi(\\psi)\\right\\|_{L^{q}_T L^{r}}& \\le\n  \\left\\|U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^{q}_TL^{r}} \n+ C\\left\\|U_V(\\cdot) \\tilde\n  \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta \n  }\\|\\psi\\|_{L^\\infty_TH^1}^{2\\si(1-\\theta) } \\|\\psi\\|_{L^q_T L^r}.\n\\end{align*}\nFrom Strichartz estimates, $U_V(\\cdot)\\tilde \\psi_- \\in L^{q}(\\R;L^{r})$, so \n\\begin{equation*}\n  \\left\\|U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^q_TL^r}  \\to 0\\quad \\text{as }T\\to +\\infty.\n\\end{equation*}\nSince $\\theta>0$, we infer that $\\Phi$ sends $X_T$ to itself, for\n$T$ sufficiently large. \n\\smallbreak\n\nWe have also, for $\\psi_2,\\psi_1\\in X_T$:\n\\begin{align*}\n  \\left\\| \\Phi(\\psi_2)-\\Phi(\\psi_1)\\right\\|_{L^q_T L^r}&\\lesssim\n  \\max_{j=1,2}\\| \\psi_j\\|_{L^k_TL^s}^{2\\si} \\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^q_T L^r}\\\\\n&\\lesssim \\left\\|U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta\n  }\\|\\tilde \\psi_-\\|_{H^1}^{2\\si(1-\\theta) }\\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^q_T L^r}.\n\\end{align*}\nUp to choosing $T$ larger, $\\Phi$ is a contraction on $X_T$, equipped\nwith the distance\n\\begin{equation*}\n  d(\\psi_2,\\psi_1) = \\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^q_T L^r} + \\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^\\infty_T L^2},\n\\end{equation*}\nwhich makes it a Banach space (see \\cite{CazCourant}). \nTherefore, $\\Phi$\nhas a unique fixed point in $X_T$, solution to\n\\eqref{eq:duhamel-}. It follows from \\eqref{eq:A} that this solution\nhas indeed an $H^1$ regularity with \n\\begin{equation*}\n   \\|\\psi(t)-e^{-it H}\\tilde\\psi_-\\|_{H^1(\\R^d)}\\Tend t {-\n      \\infty} 0.\n\\end{equation*}\n In view\nof the global well-posedness results for the Cauchy problem associated\nto \\eqref{eq:psi} (see e.g. \\cite{CazCourant}),\nthe proposition follows.\n\\end{proof}\n\\subsection{Asymptotic completeness}\n\\label{sec:AC-quant}\n\nThere are mainly three approaches to prove asymptotic completeness for\nnonlinear Schr\\\"odinger equations (without potential). The initial\napproach (\\cite{GV79Scatt}) consists in working with a\n$\\Sigma$ regularity. This  \nmakes it possible to use the operator $x+it\\nabla$, which enjoys\nseveral nice properties, and to which an important evolution law (the\npseudo-conformal conservation law) is associated; see\nSection~\\ref{sec:class} for more details. This law provides\nimportant a priori estimates, from which asymptotic completeness\nfollows very easily the the case $\\si\\ge 2\/d$, and less easily for\nsome range of $\\si$ below $2\/d$; see e.g. \\cite{CazCourant}. \n\\smallbreak\n\nThe second historical approach relaxes the localization assumption,\nand allows \nto work in $H^1(\\R^d)$, provided that $\\si>2\/d$. It is based on\nMorawetz inequalities: asymptotic completeness is then established in\n\\cite{LiSt78,GV85} for the case $d\\ge 3$, and in \\cite{NakanishiJFA} for the low\ndimension cases $d=1,2$, by introducing more intricate Morawetz\nestimates. Note that the case $d\\le 2$ is already left out in our case, since we\nhave assumed $d\\ge 3$ to prove Proposition~\\ref{prop:waveop-quant}.  \n\\smallbreak\n\nThe most recent approach to prove asymptotic completeness in $H^1$\nrelies on the introduction of interaction Morawetz estimates in \\cite{CKSTTCPAM},\nan approach which has been revisited since, in particular in\n\\cite{PlVe09} and \\cite{GiVe10}. See also \\cite{Vi09} for a very nice\nalternative approach of the use of interaction Morawetz estimates.  In\nthe presence of an external potential, this approach was used in\n\\cite{CaDa-p}, by working with Morrey-Campanato type norms. \n\\smallbreak\n\nAn analogue for the pseudo-conformal evolution law is available (see\ne.g. \\cite {CazCourant}), but it seems that in the presence of $V$\nsatisfying Assumption~\\ref{hyp:V}, it cannot be exploited to get\nsatisfactory estimates. We shall rather consider\nMorawetz estimates as in \\cite{GV85}, and thus give an alternative\nproof of the corresponding result from \\cite{CaDa-p}: note that for $\\l=1$,\nthe first part of \\eqref{eq:morawetz} provides exactly the same a\npriori estimate as in \\cite{GV85}. \n\\begin{proposition}\\label{prop:AC-quant}\n  Let $d\\ge 3$, $\\frac{2}{d}<\\si<\\frac{2}{d-2}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. There exists $M=M(\\mu,d)$ such\n  that if the attractive part of the potential satisfies\n  \\begin{equation*}\n    (\\d_r V(x))_+\\le \\frac{M}{(1+|x|)^{\\mu+1}},\\quad \\forall x\\in\n    \\R^d,\n  \\end{equation*}\nthen  for\n  all $\\varphi\\in H^1(\\R^d)$, there exist a unique $\\psi\\in\n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi} with $\\psi_{\\mid\n    t=0}=\\varphi$, and a unique $\\tilde\\psi_+\\in\n  H^1(\\R^d)$ such that\n  \\begin{equation*}\n    \\|\\psi(t)-e^{-itH}\\tilde \\psi_+\\|_{H^1(\\R^d)}\\Tend t {+\n      \\infty} 0.\n  \\end{equation*}\nIn addition, $\\psi,\\nabla \\psi\\in L^q(\\R_+,L^r(\\R^d))$ for all\nadmissible pairs $(q,r)$. \n\\end{proposition}\n\\begin{proof}\n  The proof follows that argument presented in \\cite{GV85} (and\n  resumed in \\cite{GinibreDEA}), so we shall only described the main\n  steps and the modifications needed in the present context.  The key\n  property in the proof consists in showing that there exists\n  $2<r<\\frac{2d}{d-2}$ such that \n  \\begin{equation}\\label{eq:Lp0}\n    \\|\\psi(t)\\|_{L^r}\\Tend t {+\\infty} 0.\n  \\end{equation}\nSince $\\psi\\in L^\\infty(\\R;H^1)$ (see e.g. \\cite{CazCourant}), we\ninfer that the above property is true for \\emph{all}\n$2<r<\\frac{2d}{d-2}$. This aspect is the only one that requires some\nadaptation in our case. Indeed, once this property is at hand, the end\nof the proof relies on Strichartz estimates applied to Duhamel's\nformula. In our framework, since we first want to get rid of the\nnonlinearity only (and not the potential $V$ yet), we consider\n\\begin{equation*}\n  \\psi(t) = U_V(t)\\varphi-i\\int_0^t U_V(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds,\n\\end{equation*}\nand thanks to Proposition~\\ref{prop:StrichartzRS}, it is possible to\nfollow exactly the same lines as in \\cite{GV85} (see also \\cite{TzVi-p})\nin order to infer Proposition~\\ref{prop:AC-quant}. \n\\smallbreak\n\nTherefore, the only delicate point is to show that \\eqref{eq:Lp0}\nholds for some $2<r<\\frac{2d}{d-2}$. This corresponds to Corollary~5.1\nin \\cite{GV85} (Lemme~12.6 in \\cite{GinibreDEA}). The main technical\nremark is that once Morawetz estimate is available (the one given in\nProposition~\\ref{prop:Morawetz}, whose final conclusion does not depend on\n$V$), one uses dispersive properties of the group $U(t)$. As mentioned\nabove, we do not want to use dispersive properties of $U_V(t)$, since\nthey are known only in the case $d=3$ (on the other hand, this means\nthat the result is straightforward in the case $d=3$, from \\cite{GV85}\nand \\cite{Go06}). So instead, we consider\nDuhamel's formula for \\eqref{eq:psi} in terms of $U(t)$, which reads\n\\begin{equation}\n  \\label{eq:DuhamelU}\n  \\psi(t)= U(t)\\varphi-i\\int_0^t\n  U(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds-i\\int_0^t\n  U(t-s)\\(V\\psi(s)\\)ds. \n\\end{equation}\nThe new term compared to \\cite{GV85} is of course the last term in\n\\eqref{eq:DuhamelU}, and so the nonlinearity is now\n\\begin{equation*}\n  f(\\psi) = |\\psi|^{2\\si}\\psi +V\\psi. \n\\end{equation*}\nFollowing the argument from \\cite{GV85} (or \\cite{GinibreDEA}), it\nsuffices to prove the following two properties: \\\\\n\n\\noindent 1. There exist\n$r_1>2^*=\\frac{2d}{d-2}$ and $\\alpha>0$ \nsuch that \n\\begin{equation}\n  \\left\\|\\int_{t_0}^{t-\\ell}\n  U(t-s)\\(V\\psi(s)\\)ds\\right\\|_{L^{r_1}(\\R^d)}\\le C\n\\ell^{-\\alpha}\\|\\psi\\|_{L^\\infty(\\R;H^1)}, \n\\end{equation}\nConsider a Lebesgue index $r_1$\nslightly larger than \n$2^*$, \n\\begin{equation*}\n  \\frac{1}{r_1} = \\frac{1}{2^*} -\\eta,\\quad 0<\\eta\\ll 1. \n\\end{equation*}\nLet $\\ell>0$, and consider\n\\begin{equation*}\n  I_1(t) = \\left\\|\\int_{t_0}^{t-\\ell}\n  U(t-s)\\(V\\psi(s)\\)ds\\right\\|_{L^{r_1}(\\R^d)}.\n\\end{equation*}\nStandard dispersive estimates for $U$ yield\n\\begin{equation*}\n  I_1(t) \\lesssim \\int_{t_0}^{t-\\ell} (t-s)^{-\\delta_1} \\|V\\psi(s)\\|_{L^{r'_1}}ds,\n\\end{equation*}\nwhere $\\delta_1$ is given by\n\\begin{equation*}\n  \\delta_1 = d\\(\\frac{1}{2}-\\frac{1}{r_1}\\) = 1+\\eta d.\n\\end{equation*}\nNow we apply H\\\"older inequality in space, in view of the identity\n\\begin{equation*}\n  \\frac{1}{r'_1} = \\frac{1}{2}+\\frac{1}{d}-\\eta =\n  \\underbrace{\\frac{1}{2}-\\frac{1}{d} +\\eta}_{1\/k} +\n  \\underbrace{\\frac{2}{d} -2\\eta}_{1\/q}. \n\\end{equation*}\nFor $\\eta>0$ sufficiently small, $V\\in L^q(\\R^d)$ since $\\mu>2$, and so\n\\begin{equation*}\n  \\|V\\psi(s)\\|_{L^{r'_1}} \\le \\|V\\|_{L^{q}}\\|\\psi(s)\\|_{L^k}\\lesssim \\|\\psi\\|_{L^\\infty(\\R;H^1)},\n\\end{equation*}\nwhere we have used Sobolev embedding, since $2<k<2^*$. We infer\n\\begin{align*}\n  I_1(t) &\\lesssim  \\int_{t_0}^{t-\\ell} (t-s)^{-\\delta_1} ds\n  \\|\\psi\\|_{L^\\infty(\\R;H^1)} \\lesssim  \\int_{\\ell}^{\\infty}\n  s^{-\\delta_1} ds\\|\\psi\\|_{L^\\infty(\\R;H^1)}\\\\\n& \\lesssim\n  \\ell^{1-\\delta_1}\\|\\psi\\|_{L^\\infty(\\R;H^1)}=\\ell^{-\\eta d}\\|\\psi\\|_{L^\\infty(\\R;H^1)}. \n\\end{align*}\n\n\\noindent 2. Now for fixed $\\ell>0$, let \n\\begin{equation*}\n  I_2(t) = \\left\\|\\int_{t-\\ell}^t\n  U(t-s)\\(V\\psi(s)\\)ds\\right\\|_{L^{2\\si+2}(\\R^d)}.\n\\end{equation*}\nWe show that for any  $\\ell>0$, $I_2(t)\\to 0$ as $t\\to \\infty$. \nDispersive estimates for $U(t)$ yield\n\\begin{equation*}\n  I_2(t) \\lesssim \\int_{t-\\ell}^t\n  (t-s)^{-\\delta}\\|V\\psi(s)\\|_{L^{\\frac{2\\si+2}{2\\si+1}}}ds,\\quad\n  \\delta = d\\(\\frac{1}{2}-\\frac{1}{2\\si+2}\\) = \\frac{d\\si}{2\\si+2}<1. \n\\end{equation*}\nFor (a small) $\\alpha$ to be fixed later, H\\\"older inequality yields\n\\begin{equation*}\n  \\|V\\psi(s)\\|_{L^{\\frac{2\\si+2}{2\\si+1}}} =\\left\\| |x|^{\\alpha}V\n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{\\frac{2\\si+2}{2\\si+1}}}\n  \\le \\left\\| |x|^{\\alpha}V \\right\\|_{L^{\\frac{\\si+1}{\\si}}}\n\\left\\| \n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{2\\si+2}}.\n\\end{equation*}\nNote that for $0<\\alpha\\ll 1$, $\\left\\| |x|^{\\alpha}V\n\\right\\|_{L^{\\frac{\\si+1}{\\si}}}$ is finite, since\n$\\frac{\\si+1}{\\si}>\\frac{d}{2}$ and $\\mu>2$. For $0<\\theta<1$, write \n\\begin{align*}\n  \\left\\| \n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{2\\si+2}}=  \\left\\| \n    \\frac{|\\psi(s)|^{\\theta}}{|x|^\\alpha}\n      |\\psi(s)|^{1-\\theta}\\right\\|_{L^{2\\si+2}}&\\le \\left\\| \n    \\frac{\\psi(s)}{|x|^{\\alpha\/\\theta}}\\right\\|_{L^{2\\si+2}}^\\theta\n      \\left\\|\\psi(s)\\right\\|_{L^{2\\si+2}}^{1-\\theta}\\\\\n   &   \\lesssim  \\left\\| \n    \\frac{\\psi(s)}{|x|^{\\alpha\/\\theta}}\\right\\|_{L^{2\\si+2}}^\\theta\n      \\left\\|\\psi\\right\\|_{L^\\infty(\\R;H^1)}^{1-\\theta}.\n\\end{align*}\nTo use Morawetz estimate, we impose $\\alpha\/\\theta= 1\/(2\\si+2)$, so\nthat we have\n\\begin{equation*}\n   \\left\\| \n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{2\\si+2}} \\lesssim\n\\(  \\int_{\\R^d}\\frac{|\\psi(s,x)|^{2\\si+2}}{|x|}dx\\)^{\\theta\/(2\\si+2)}\n\\left\\|\\psi\\right\\|_{L^\\infty(\\R;H^1)}^{1-\\theta}. \n\\end{equation*}\nWe conclude by applying H\\\"older inequality in time: since $\\delta<1$,\nthe map $s\\mapsto (t-s)^{-\\delta}$ belongs to $L^q_{\\rm loc}$ for $1\\le\n  q\\le 1+\\gamma$ and $\\gamma>0$ sufficiently small. Let $q=1+\\gamma$\n  with $0<\\gamma\\ll 1$ so that $s\\mapsto (t-s)^{-\\delta}\\in L^q_{\\rm\n    loc}$: we have $q'<\\infty$, and we can choose  $0<\\theta\\ll 1$ (or\n  equivalently $0<\\eta\\ll 1$) so\n  that \n  \\begin{equation*}\n    \\theta q'=2\\si+2. \n  \\end{equation*}\nWe end up with \n\\begin{equation*}\n I_2(t) \\lesssim \\ell^\\beta \\(\\iint_{[t-\\ell,t]\\times\\R^d}\n   \\frac{|\\psi(s,x)|^{2\\si+2}}{|x|}dsdx\\)^{1\/(2\\si+2)q'},\n\\end{equation*}\nfor some $\\beta>0$. The last factor goes to zero as $t\\to \\infty$ from\nProposition~\\ref{prop:Morawetz}. \n\\end{proof}\n\n\n\\subsection{Scattering}\n\\label{sec:conclusion}\n\nUnder Assumption~\\ref{hyp:V}, a linear scattering theory is available,\nprovided that $\\mu>1$; see e.g. \\cite[Section~4.6]{DG}. This means that\nthe following strong limits exist in $L^2(\\R^d)$,\n\\begin{equation*}\n  \\lim_{t\\to -\\infty} U_V(-t)U(t),\\quad\\text{and}\\quad  \\lim_{t\\to +\\infty} U(-t)U_V(t),\n\\end{equation*}\nwhere the second limit usually requires to project on the continuous\nspectrum. Recall that this projection is the identity in our\nframework. \n\\begin{lemma}\\label{lem:Cook-quant}\n  Let $d\\ge 3$, $V$ satisfying Assumption~\\ref{hyp:V} with $p>1$. Then \nthe strong limit\n \\begin{equation*}\n  \\lim_{t\\to -\\infty} U_V(-t)U(t)\n\\end{equation*}\nexists in $H^1(\\R^d)$. \n\\end{lemma}\n\\begin{proof}\n  Following Cook's method (\\cite[Theorem~XI.4]{ReedSimon3}), it\n  suffices to prove that for all $\\varphi \\in \\Sch(\\R^d)$,\n  \\begin{equation*}\n    t\\mapsto \\left\\| U_V(-t) VU(t)\\varphi\\right\\|_{H^1}\\in\n    L^1((-\\infty,-1]). \n  \\end{equation*}\nFor the $L^2$ norm, we have\n\\begin{equation*}\n  \\left\\| U_V(-t) VU(t)\\varphi\\right\\|_{L^2} = \\left\\|\n    VU(t)\\varphi\\right\\|_{L^2} .\n\\end{equation*}\nAssumption~\\ref{hyp:V} implies that $V\\in L^q(\\R^d)$ for all\n$q>d\/\\mu$. For $\\mu>1$, let $q$ be given by \n\\begin{equation*}\n  \\frac{1}{q} = \\frac{1}{d}+\\eta,\\text{ with } \\eta>0\\text{ and }q>\\frac{d}{\\mu}. \n\\end{equation*}\nWe apply H\\\"older inequality with the identity\n\\begin{equation*}\n  \\frac{1}{2} = \\frac{1}{q} +\\underbrace{\\frac{1}{2}-\\frac{1}{d}-\\eta}_{1\/r}.\n\\end{equation*}\nUsing dispersive estimates for $U(t)$, we have\n\\begin{equation*}\n  \\left\\|\n    VU(t)\\varphi\\right\\|_{L^2} \\lesssim \\|U(t)\\varphi\\|_{L^r}\\lesssim\n  |t|^{-d\\(\\frac{1}{2}-\\frac{1}{r}\\)}\\|\\varphi\\|_{L^{r'}}= |t|^{-1-d\\eta}\\|\\varphi\\|_{L^{r'}},\n\\end{equation*}\nhence the existence of the strong limit in $L^2$. \n\\smallbreak\n\nFor the $H^1$ limit, recall that from Lemma~\\ref{lem:A}, \n\\begin{equation*}\n  \\left\\| \\nabla U_V(-t) VU(t)\\varphi\\right\\|_{L^2}\\lesssim \\left\\| A\n    U_V(-t) VU(t)\\varphi\\right\\|_{L^2} \n\\end{equation*}\nSince $A$ commutes with $U_V$ which is unitary on $L^2$, the right\nhand side is equal to \n\\begin{equation*}\n\\left\\| A\n    VU(t)\\varphi\\right\\|_{L^2}\\lesssim \\|VU(t)\\varphi\\|_{H^1},\n\\end{equation*}\nwhere we have used Lemma~\\ref{lem:A} again. Now\n\\begin{equation*}\n  \\|VU(t)\\varphi\\|_{H^1} \\le \\|VU(t)\\varphi\\|_{L^2}+ \\|\\nabla V\\times\n  U(t)\\varphi\\|_{L^2} + \\|VU(t)\\nabla \\varphi\\|_{L^2},\n\\end{equation*}\nand each term is integrable, like for the $L^2$ limit, from\nAssumption~\\ref{hyp:V}. \n\\end{proof}\n\n  In the case $d=3$, the dispersive estimates established by Goldberg\n  \\cite{Go06} make it possible to prove asymptotic completeness in\n  $H^1$ by Cook's method as well: for all $\\varphi\\in\n  \\Sch(\\R^d)$,\n  \\begin{equation*}\n    t\\mapsto \\|U(-t)VU_V(t)\\varphi\\|_{H^1}\\in L^1(\\R),\n  \\end{equation*}\na property which can be proven by the same computations as above, up\nto changing the order of the arguments. To complete the proof of\nTheorem~\\ref{theo:scatt-quant}, it therefore remains to prove that for\n$d\\ge 4$, $\\psi_+\\in H^1(\\R^d)$ and\n  \\begin{equation}\\label{eq:cvH1}\n    \\|\\psi(t)-U(t)\\psi_+\\|_{H^1(\\R^d)}\\Tend t \\infty 0. \n  \\end{equation}\nIt follows from the above results that\n\\begin{equation*}\n  \\psi(t) = U(t) \\psi_+ +i\\int_t^{+\\infty}\n  U(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds\n  +i\\int_t^{+\\infty}U(t-s)\\(V(\\psi(s)\\)ds,\n\\end{equation*}\nand that $\\psi,\\nabla \\psi \\in L^q(\\R;L^r(\\R^d))$ for all admissible\npairs $(q,r)$. Since we have\n\\begin{equation*}\n  \\psi_+= U(-t)\\psi(t) - i\\int_t^{+\\infty}\n  U(-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds\n -i\\int_t^{+\\infty}U(-s)\\(V(\\psi(s)\\)ds,\n\\end{equation*}\nthe previous estimates show that $\\psi_+\\in H^1(\\R^d)$, along with\n\\eqref{eq:cvH1}. \n\n\n\n\\section{Scattering for the asymptotic envelope}\n\\label{sec:class}\n\n\nIn this section, we prove Theorem~\\ref{theo:scatt-class}. The general\nargument is similar to the quantum case: we first prove that the\nnonlinear term can be neglected to large time, and then rely on\nprevious results to neglect the potential. \nRecall that in view of Assumption~\\ref{hyp:V}, the time dependent\nharmonic potential $\\frac{1}{2}\\<Q(t)y,y\\>$ satisfies\n \\begin{equation}\\label{eq:decayQ}\n   \\left\\|\\frac{d^\\alpha}{dt^\\alpha}Q(t)\\right\\|\\lesssim\n   \\<t\\>^{-\\mu-2-\\alpha},\\quad \\alpha \\in \\N,\n \\end{equation}\nwhere $\\|\\cdot\\|$ denotes any matricial norm. \nWe denote by \n\\begin{equation*}\n  H_Q = -\\frac{1}{2}\\Delta + \\frac{1}{2}\\<Q(t)y,y\\>\n\\end{equation*}\nthe time-dependent Hamiltonian present in \\eqref{eq:u}. Like in the\nquantum case, we show that the nonlinearity is negligible for large\ntime by working on Duhamel's formula associated to \\eqref{eq:u} in\nterms of $H_Q$. Since $H_Q$ depends on time, we recall that the\npropagator $U_Q(t,s)$ is the operator which maps $u_0$ to $u_{\\rm lin}(t)$,\nwhere $u_{\\rm lin}$ solves\n\\begin{equation*}\n  i\\d_t u_{\\rm lin} +\\frac{1}{2}\\Delta u_{\\rm lin} =\n  \\frac{1}{2}\\<Q(t)y,y\\>u_{\\rm lin};\\quad u_{{\\rm lin}}(s,y)=u_0(y). \n\\end{equation*}\nIt is a unitary dynamics, in the sense that $U_Q(s,s)=1$, and\n$U_Q(t,\\tau)U_Q(\\tau,s)=U_Q(t,s)$;\nsee e.g. \\cite{DG}. Then to prove the existence of wave operators, we consider the\nintegral formulation\n\\begin{equation}\n  \\label{eq:duhamel-wave-class}\n  u(t) = U_Q(t,0)\\tilde u_--i\\int_{-\\infty}^t U_Q(t,s)\\(|u|^{2\\si}u(s)\\)ds.\n\\end{equation}\nA convenient tool is given by Strichartz estimates associated to\n$U_Q$. Local in time Strichartz estimates follow from general results\ngiven in \\cite{Fujiwara}, where local dispersive estimates are\nproven for more general potential. To address large time,  we take\nadvantage of the fact that the \npotential is exactly quadratic with respect to the space variable, so\nan explicit formula is available for $U_Q$, entering the general\nfamily of Mehler's formulas (see e.g. \\cite{Feyn,HormanderQuad}). \n\\subsection{Mehler's formula}\n\\label{sec:mehler}\n\nConsider, for $t_0\\ll -1$,\n\\begin{equation*}\ni\\d_tu+\\frac{1}{2}\\Delta u=\\frac{1}{2}\\< Q(t)y,y\\> u\\quad\n;\\quad u(t_0,y)=u_0(y).\n\\end{equation*}\nWe seek a solution of the form\n\\begin{equation}\n  \\label{eq:mehler}\n  u(t,y) = \\frac{1}{h(t)}\\int_{\\R^d}\n  e^{\\frac{i}{2}\\(\\<M_1(t)y,y\\>+\\<M_2(t)z,z\\>+2\\<P(t)y,z\\>\\)}u_0(z)dz, \n\\end{equation}\nwith symmetric matrices $M_1, M_2,P\\in \\mathcal S_d(\\R)$. \nExperience shows that no linear term is needed in this formula, since\nthe potential is exactly quadratic (see\ne.g. \\cite{CLSS08}). \n\\smallbreak\n\nWe compute:\n\\begin{align*}\n  i\\d_t u & = -i\\frac{\\dot h}{h}u -\\frac{1}{2}\\<\\dot M_1(t)y,y\\>u\\\\\n&\\quad \n  +\\frac{1}{h}\\int e^{\\frac{i}{2}\\(\\dots\\)} \\(-\\frac{1}{2}\\<\\dot\n  M_2(t)z,z\\>-\\<\\dot P(t)y,z\\>\\)u_0(z)dz,\n\\end{align*}\n\\begin{align*}\n  \\d_{j}^2 u &= \\frac{1}{h}\\int e^{\\frac{i}{2}\\(\\dots\\)}\n  \\(-\\(\\(M_1(t)y\\)_j + \\(P(t)z\\)_j\\)^2 -i\\(M_1\\)_{jj}\\)u_0(z)dz,\n\\end{align*}\nhence\n\\begin{align*}\n & i\\d_tu+\\frac{1}{2}\\Delta u =  -i\\frac{\\dot h}{h}u\n  +\\frac{i}{2}\\operatorname{tr} M_1 - \\frac{1}{2}\\<\\dot M_1(t)y,y\\>u\\\\\n&+ \\frac{1}{2h}\\int\n  e^{\\frac{i}{2}\\(\\<M_1(t)y,y\\>+\\<M_2(t)z,z\\>+2\\<P(t)y,z\\>\\)}u_0(z)\\times\\\\\n&\\times\\( \n-\\<\\dot  M_2(t)z,z\\>-2\\<\\dot\nP(t)y,z\\>-|M_1(t)y|^2 -|P(t)z|^2 -2\n\\<M_1(t)y,P(t)z\\>\\)dz. \n\\end{align*}\nIdentifying the quadratic forms (recall that the matrices $M_j$ and\n$P$ are symmetric), we find:\n\\begin{align*}\n&\\frac{\\dot h}{h}=  \\frac{1}{2}\\operatorname{tr} M_1,\\\\\n& \\dot M_1+M_1^2+Q=0,\\\\\n& \\dot M_2 +P^2=0,\\\\\n& \\dot P + PM_1=0.\n\\end{align*}\nDispersion is given by\n\\begin{equation*}\n  h(t)  = h(t_1)\\exp\\(\\frac{1}{2}\\int_{t_1}^t \\operatorname{tr} M_1(s)ds\\),\n\\end{equation*}\nwhere $M_1$ solves the matrix Riccati equation\n\\begin{equation}\n  \\label{eq:riccati}\n   \\dot M_1 + M_1^2 + Q=0;\\quad M_1(t_0)=\\frac{1}{t_0}{\\rm I}_d.\n\\end{equation}\nNote that in general, solutions to Riccati equations develop\nsingularities in finite time. What saves the day here is that\n\\eqref{eq:riccati} is not translation invariant, and can be\nconsidered, for $t\\le t_0\\ll -1$, \nas a perturbation of the Cauchy problem\n\\begin{equation*}\n  \\dot M + M^2 =0;\\quad M(t_0)=\\frac{1}{t_0}{\\rm I}_d,\n\\end{equation*}\nwhose solution is given by \n\\begin{equation*}\n  M(t) = \\frac{1}{t}{\\rm I}_d. \n\\end{equation*}\n\\begin{lemma}\\label{lem:riccati}\n  Let $Q$ be a symmetric matrix satisfying \\eqref{eq:decayQ} for $\\mu>1$. There\n  exists $t_0<0$ such that \\eqref{eq:riccati} has a unique solution\n  $M_1\\in C((-\\infty,t_0];\\mathcal S_d(\\R))$. In addition, it\n  satisfies\n  \\begin{equation*}\n    M_1(t)= \\frac{1}{t}{\\rm I}_d +\\O\\(\\frac{1}{t^2}\\)\\quad \\text{as\n    }t\\to -\\infty. \n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSeek a solution of the form $M_1(t) =   \\frac{1}{t}{\\rm I}_d +R(t)$,\nwhere $R$ is s symmetric matrix solution of\n\\begin{equation*}\n  \\dot R + \\frac{2}{t}R+R^2 +Q= 0;\\quad R(t_0)=0. \n\\end{equation*}\nEquivalently, the new unknown $\\tilde R = t^2 R$ must satisfy\n\\begin{equation}\\label{eq:Rmatrix}\n   \\dot {\\tilde R} + \\frac{1}{t^2}\\tilde R^2 +t^2Q= 0;\\quad \\tilde R(t_0)=0. \n\\end{equation}\nCauchy-Lipschitz Theorem yields a local solution: we show that it is\ndefined on $(-\\infty,t_0]$, along with the announced decay. \nIntegrating between $t_0$ and $t$, we find\n\\begin{equation*}\n  \\tilde R(t) = -\\int_{t_0}^t \\frac{1}{s^2}\\tilde R(s)^2ds -\n  \\int_{t_0}^ts^2 Q(s)ds.\n\\end{equation*}\nNote that $s\\mapsto s^2 Q$ is integrable as $s\\to -\\infty$ from \\eqref{eq:decayQ}\n(we assume $\\mu>1$). Setting \n\\begin{equation*}\n  \\rho(t) =\\sup_{t\\le s\\le t_0}\\|\\tilde R(s)\\|,\n\\end{equation*}\nwhere $\\|\\cdot\\|$ denotes any matricial norm, we have\n\\begin{equation*}\n  \\rho(t) \\le \\frac{C}{t_0}\\rho(t)^2 + \\frac{C}{t_0^{\\mu-1}},\n\\end{equation*}\nfor some constant $C$. Choosing $t_0\\ll -1$, global existence follows\nfrom the following bootstrap argument (see \\cite{BG3}):\nLet $f=f(t)$ be a nonnegative continuous function on $[0,T]$ such\nthat, for every $t\\in [0,T]$, \n\\begin{equation*}\n  f(t)\\le \\eps_1 + \\eps_2 f(t)^\\theta,\n\\end{equation*}\nwhere $\\eps_1,\\eps_2>0$ and $\\theta >1$ are constants such that\n\\begin{equation*}\n  \\eps_1 <\\left(1-\\frac{1}{\\theta} \\right)\\frac{1}{(\\theta \\eps_2)^{1\/(\\theta\n-1)}}\\ ,\\ \\ \\ f(0)\\le  \\frac{1}{(\\theta \\eps_2)^{1\/(\\theta-1)}}.\n\\end{equation*}\nThen, for every $t\\in [0,T]$, we have\n\\begin{equation*}\n  f(t)\\le \\frac{\\theta}{\\theta -1}\\ \\eps_1.\n\\end{equation*}\nThis shows that for $|t_0|$ sufficiently large, the matrix $R$ (hence\n$M_1$) is defined on $(-\\infty,t_0]$. Moreover, since $\\tilde R$ is\nbounded, $R(t)=\\O(t^{-2})$ as $t\\to -\\infty$, hence the result. \n\\end{proof}\nWe infer\n\\begin{equation*}\n  h(t)\\Eq t {-\\infty} c|t|^{d\/2},\n\\end{equation*}\nwhich is the same dispersion as in the case without\npotential. Putting this result together with local dispersive estimates from\n\\cite{Fujiwara}, we have:\n\\begin{lemma}\\label{lem:strichartz-quad}\n  Let $Q$ be a symmetric matrix satisfying \\eqref{eq:decayQ} for\n  $\\mu>1$. Then for all admissible pairs $(q,r)$, \nthere exists $C=C(q,d)$ such that for all $s\\in \\R$,\n\\begin{equation*}\n  \\|U_Q(\\cdot,s)f\\|_{L^q(\\R;L^r(\\R^d))}\\le C \\|f\\|_{L^2(\\R^d)},\\quad\n  \\forall f\\in L^2(\\R^d). \n\\end{equation*}\nFor two admissible pairs $(q_1,r_1)$ and $(q_2,r_2)$, there exists $C_{q_1,q_2}$ such\nthat for all time interval $I$, if we denote by\n\\begin{equation*}\n  R(F)(t,y) = \\int_{I\\cap \\{s\\le t\\}} U_Q(t,s)F(s,y)ds,\n\\end{equation*}\nwe have\n\\begin{equation*}\n  \\|R(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^d))}\\le\n  C_{q_1,q_2}\\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^d))},\\quad \\forall F\\in\n  L^{q_2'}(I;L^{r_2'}(\\R^d)). \n\\end{equation*}\n\\end{lemma}\n\\begin{remark}\n  Since we have dispersive estimates, end-point Strichartz estimates\n  ($q=2$ when $d\\ge 3$)\n  are also available from \\cite{KT}. \n\\end{remark}\n\\subsection{Wave operators}\n\\label{sec:wave-class}\n\nIn this section, we prove:\n\\begin{proposition}\\label{prop:wave-class}\n   Let $d\\ge 1$, $\\frac{2}{d}\\le \\si<\\frac{2}{(d-2)_+}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>1$. For\n  all $\\tilde u_-\\in \\Sigma$, there exists a unique $u\\in\n  C(\\R;\\Sigma)$ solution to \\eqref{eq:u}  such that\n  \\begin{equation*}\n    \\|U_Q(0,t)u(t)-\\tilde u_-\\|_{\\Sigma}\\Tend t {-\n      \\infty} 0.\n  \\end{equation*}\n\\end{proposition}\n\\begin{remark}\n  The assumption $\\si\\ge \\frac{2}{d}$ could easily be relaxed,\n  following the classical argument (see e.g. \\cite{CazCourant}). We do\n  not present the argument, since Theorem~\\ref{theo:scatt-quant} is\n  proven only for $\\si>\\frac{2}{d}$. \n\\end{remark}\n\n\\begin{proof}\n  The proof follows closely the approach without potential\n  ($Q=0$). From this perspective, a key tool is the vector field\n  \\begin{equation*}\n    J(t)=y+it\\nabla.\n  \\end{equation*}\nIt satisfies three important properties:\n\\begin{itemize}\n\\item It commutes with the free Schr\\\"odinger dynamics,\n  \\begin{equation*}\n   \\left[ i\\d_t +\\frac{1}{2}\\Delta,J\\right]=0.  \n  \\end{equation*}\n\\item It acts like a derivative on gauge invariant nonlinearities. If\n  $F(z)$ is of the form $F(z)=G(|z|^2)z$, then \n  \\begin{equation*}\n    J(t)\\(F(u)\\) = \\d_z F(u)J(t)u -\\d_{\\bar z}F(u)\\overline{J(t)u}.\n  \\end{equation*}\n\\item It provides weighted Gagliardo-Nirenberg inequalities:\n  \\begin{align*}\n    \\|f\\|_{L^r}\\lesssim &\n    \\frac{1}{|t|^{\\delta(r)}}\\|f\\|_{L^2}^{1-\\delta(r)}\\|J(t)f\\|_{L^2}^{\\delta(r)},\n    \\quad \\delta(r)=d\\(\\frac{1}{2}-\\frac{1}{r}\\), \\\\\n&\\text{with }\n\\left\\{\n  \\begin{aligned}\n    2\\le r\\le \\infty &\\text{ if }d=1,\\\\\n2\\le r<\\infty &\\text{ if }d=2,\\\\\n2\\le r\\le \\frac{2d}{d-2}&\\text{ if }d\\ge 3. \n  \\end{aligned}\n\\right.\n  \\end{align*}\n\\end{itemize}\nThe last two properties stem from the factorization $J(t)f =\nit e^{i\\frac{|y|^2}{2t}}\\nabla \\(e^{-i\\frac{|y|^2}{2t}}f\\)$. Note that\nthe commutation property does not incorporate the quadratic potential:\n\\begin{align*}\n  \\left[ i\\d_t -H_Q,J\\right]= itQ(t)y=itQ(t)J(t) +t^2 Q(t)\\nabla. \n\\end{align*}\nNow the important remark is that $t\\mapsto t^2Q(t)$ is integrable,\nfrom \\eqref{eq:decayQ} since $\\mu>1$.  \n\\smallbreak\n\nTo prove Proposition~\\ref{prop:wave-class}, we apply a fixed point argument \nto the Duhamel's formula \\eqref{eq:duhamel-wave-class}. As in the case\nof the quantum scattering operator, we have to deal with the fact that\nthe gradient does not commute with $U_Q$, leading to the problem\ndescribed in Section~\\ref{sec:vector-field}. Above, we have sketched\nhow to deal with the inhomogeneous term in\n\\eqref{eq:duhamel-wave-class}, while in\nSection~\\ref{sec:vector-field}, we had underscored the difficulty\nrelated to the homogeneous term. We therefore start by showing that\nfor any admissible pair $(q_1,r_1)$, there exists $K_{q_1}$ such that\n\\begin{equation}\\label{eq:Qhomo}\n  \\|\\nabla U_Q(t,0)f\\|_{L^{q_1}(\\R;L^{r_1})} +\n  \\|J(t)U_Q(t,0)f\\|_{L^{q_1}(\\R;L^{r_1})} \\le K_{q_1} \\|f\\|_{\\Sigma}. \n\\end{equation}\nTo prove this, denote \n\\begin{equation*}\n  v_0(t)=U_Q(t,0)f,\\quad v_1(t)= \\nabla U_Q(t,0)f, \\quad v_2(t)=\nJ(t)U_Q(t,0)f.\n\\end{equation*}\nSince $yv_0 =v_2-it v_1$, we have:\n\\begin{align*}\n  &i\\d_t v_1=H_Q v_1 +Q(t)yv_0 = Hv_1 +Q(t)v_2-it Q(t)v_1;\\quad\n    v_1(0,y)=\\nabla f(y),\\\\\n& i\\d_t v_2 = H_Qv_2 +itQ(t)v_2+t^2Q(t)v_1;\\quad v_2(0,y)=yf(y). \n\\end{align*}\nLemma~\\ref{lem:strichartz-quad} yields\n\\begin{align*}\n  \\|v_1\\|_{L^{q_1}(\\R;L^{r_1})} + \\|v_2\\|_{L^{q_1}(\\R;L^{r_1})}\n  &\\lesssim \\|f\\|_\\Sigma + \\int_{-\\infty}^\\infty \\|\\<t\\>Q(t)v_2(t)\\|_{L^2}dt \\\\\n&\\quad +\n  \\int_{-\\infty}^\\infty \\|\\<t\\>^2Q(t)v_1(t)\\|_{L^2}dt ,\n\\end{align*}\nwhere we have chosen $(q_2,r_2)=(\\infty,2)$. The fact that $U_Q$ is\nunitary on $L^2$ and \\eqref{eq:decayQ} imply\n\\begin{equation*}\n  \\|\\<t\\>Q(t)v_2(t)\\|_{L^2}\\lesssim \\<t\\>^{-\\mu-1}\\|yf\\|_{L^2},\\quad\n  \\|\\<t\\>^2Q(t)v_1(t)\\|_{L^2}\\lesssim \\<t\\>^{-\\mu}\\|\\nabla f\\|_{L^2}, \n\\end{equation*}\nhence \\eqref{eq:Qhomo}. \nWe then apply a fixed point\nargument in\n\\begin{align*}\n  X(T) =&\\Big\\{ u\\in L^\\infty((-\\infty,-T];H^1),  \\\\\n& \\quad\\sum_{B\\in \\{{\\rm Id}, \\nabla, J\\}}\n\\( \\|B   u\\|_{L^\\infty((-\\infty,-T];L^2)}+\n\\|B  u\\|_{L^q((-\\infty,-T];L^r)}\\)\\le {\\mathbf K}\\|\\tilde u_-\\|_{\\Sigma}\\Big\\},\n\\end{align*}\nwhere the admissible pair $(q,r)$ is given by\n\\begin{equation*}\n  (q,r) = \\(\\frac{4\\si+4}{d\\si},2\\si+2\\),\n\\end{equation*}\n and the constant $\\mathbf K$ is related to the constants $C_q$  from\n Strichartz inequalities \n(Lemma~\\ref{lem:strichartz-quad}), and $K_q$  from\n\\eqref{eq:Qhomo}, whose value we do not try to optimize. The fixed\npoint argument is applied \nto the Duhamel's formula \\eqref{eq:duhamel-wave-class}: we denote by\n$\\Phi(u)$ the left hand side, and let $u\\in X(T)$. We have \n\\begin{equation*}\n  \\|\\Phi(u)\\|_{L^\\infty((-\\infty,-T];L^2)}\\le \\|\\tilde u_-\\|_{L^2} + C\n    \\left\\| |u|^{2\\si}u\\right\\|_{L^{q'}_TL^{r'}},\n\\end{equation*}\nwhere $L^a_T$ stands for $L^a((-\\infty,-T])$. H\\\"older inequality\nyields\n\\begin{equation*}\n  \\left\\| |u|^{2\\si}u\\right\\|_{L^{q'}_TL^{r'}} \\le\n  \\|u\\|_{L^k_TL^r}^{2\\si} \\|u\\|_{L^q_TL^r},\n\\end{equation*}\nwhere $k$ is given by\n\\begin{equation*}\n  \\frac{1}{q'}=\\frac{1}{q}+\\frac{2\\si}{k},\\text{ that is }k =\n  \\frac{4\\si(\\si+1)}{2-(d-2)\\si}. \n\\end{equation*}\nWeighted Gagliardo-Nirenberg inequality and the definition of $X(T)$ yield\n\\begin{equation*}\n  \\|u(t)\\|_{L^r}\\lesssim \\frac{1}{|t|^{\\frac{d\\si}{2\\si+2}}}\\|u_-\\|_{\\Sigma}.\n\\end{equation*}\nWe check that for $\\si\\ge \\frac{2}{d}$, \n\\begin{equation*}\n  k\\times \\frac{d\\si}{2\\si+2} = \\frac{2d\\si^2}{2-(d-2)\\si}\\ge 2,\n\\end{equation*}\nand so\n\\begin{equation*}\n  \\|u\\|_{L^k_TL^r}^k =\\O\\(\\frac{1}{T}\\) \\text{ as }T\\to \\infty. \n\\end{equation*}\nBy using Strichartz estimates again,\n\\begin{equation*}\n  \\|\\Phi(u)\\|_{L^q_TL^r}\\le C_q\\|\\tilde u_-\\|_{L^2} + C\n    \\left\\| |u|^{2\\si}u\\right\\|_{L^{q'}_TL^{r'}},\n\\end{equation*}\nwhich shows, like above, that if $T$ is sufficiently large,\n$\\|\\Phi(u)\\|_{L^q_TL^r}\\le 2C_q\\|\\tilde u_-\\|_{L^2}$. \n\\smallbreak\n\nWe now apply $\\nabla$ and $J(t)$ to $\\Phi$, and get a closed system of\nestimates:\n\\begin{align*}\n  \\nabla \\Phi(u) &= \\nabla U_Q(t,0)\\tilde u_- - i \\int_{-\\infty}^t\n  U_Q(t,s)\\nabla \\(|u|^{2\\si}u(s)\\)ds \\\\\n& -i\\int_{-\\infty}^t U_Q(t,s)\\(Q(s)J(s)\\Phi(u)\\)ds - \\int_{-\\infty}^t\n  U_Q(t,s)\\(sQ(s)\\nabla\\Phi(u)\\)ds, \\\\\n J(t) \\Phi(u) &= J(t) U_Q(t,0)\\tilde u_- - i \\int_{-\\infty}^t\n  U_Q(t,s)J(s) \\(|u|^{2\\si}u(s)\\)ds \\\\\n& +\\int_{-\\infty}^t U_Q(t,s)\\(sQ(s)J(s)\\Phi(u)\\)ds - i\\int_{-\\infty}^t\n  U_Q(t,s)\\(s^2Q(s)\\nabla\\Phi(u)\\)ds,\n\\end{align*}\nwhere we have used the same algebraic properties as in the proof of\n\\eqref{eq:Qhomo}. Set \n\\begin{equation*}\n  M(T) = \\sum_{B\\in \\{\\nabla, J\\}} \\( \\|B(t)\\Phi(u)\\|_{L^\\infty_TL^2}\n  + \\|B(t)\\Phi(u)\\|_{L^q_TL^r}\\).\n\\end{equation*}\nLemma~\\ref{lem:strichartz-quad} and\n\\eqref{eq:Qhomo} yield\n\\begin{align*}\n  M(T)&\\lesssim \\|\\tilde u_-\\|_\\Sigma + \\sum_{B\\in \\{\\nabla, J\\}}\\left\\||u|^{2\\si}B\n        u\\right\\|_{L^{q'}_TL^{r'}} \\\\\n&\\quad+\\|\\<t\\>Q(t)J(t)\\Phi(u)\\|_{L^1_TL^2} +\n        \\|\\<t\\>^2Q(t)\\nabla\\Phi(u)\\|_{L^1_TL^2}, \n\\end{align*}\nwhere we have also used the fact that $J(t)$ acts like a derivative on\ngauge invariant nonlinearities. The same H\\\"older inequalities as\nabove yield\n\\begin{equation*}\n  \\left\\||u|^{2\\si}B\n        u\\right\\|_{L^{q'}_TL^{r'}} \\le\n      \\|u\\|_{L^k_TL^r}^{2\\si}\\|Bu\\|_{L^q_TL^r}\\lesssim \\frac{1}{T^{2\\si\/k}}\\|Bu\\|_{L^q_TL^r}.\n\\end{equation*}\nOn the other hand, from \\eqref{eq:decayQ},\n\\begin{equation*}\n  \\|\\<t\\>Q(t)J(t)\\Phi(u)\\|_{L^1_TL^2} +\n        \\|\\<t\\>^2Q(t)\\nabla\\Phi(u)\\|_{L^1_TL^2}\\lesssim \\frac{1}{T^{\\mu-1}}M(T),\n\\end{equation*}\nand so\n\\begin{equation*}\n  M(T) \\lesssim \\|\\tilde u_-\\|_\\Sigma  +\n  \\frac{1}{T^{2\\si\/k}}\\sum_{B\\in \\{\\nabla, J\\}} \\|Bu\\|_{L^q_TL^r} + \\frac{1}{T^{\\mu-1}}M(T).\n\\end{equation*}\nBy choosing $T$ sufficiently large, we infer\n\\begin{equation*}\n  M(T) \\lesssim \\|\\tilde u_-\\|_\\Sigma  +\n  \\frac{1}{T^{2\\si\/k}}\\sum_{B\\in \\{\\nabla, J\\}} \\|Bu\\|_{L^q_TL^r},\n\\end{equation*}\nand we conclude that $\\Phi$ maps $X(T)$ to $X(T)$ for $T$\nsufficiently large. Up to choosing $T$ even larger, $\\Phi$ is a\ncontraction on $X(T)$ with respect to the weaker norm $L^q_TL^r$,\nsince for $u,v\\in X(T)$, we have \n\\begin{align*}\n  \\|\\Phi(u)-\\Phi(v)\\|_{L^q_TL^r}&\\lesssim \\left\\| |u|^{2\\si}u\n    -|v|^{2\\si}v\\right\\|_{L^{q'}_TL^{r'}}\\lesssim \\(\n  \\|u\\|_{L^k_TL^r}^{2\\si}+\\|v\\|_{L^k_TL^r}^{2\\si}\\)\\|u-v\\|_{L^q_TL^r}\\\\\n&\\lesssim \\frac{1}{T^{2\\si\/k}}\\|u-v\\|_{L^q_TL^r},\n\\end{align*}\nwhere we have used the previous estimate. Therefore, there exists\n$T>0$ such that $\\Phi$ has a unique fixed point in $X(T)$. This\nsolution actually belongs to $C(\\R;\\Sigma)$ from \\cite{CaSi15}.  \nUnconditional uniqueness (in $\\Sigma$, without referring to mixed\nspace-time norms) stems from the approach in \\cite{TzVi-p}. \n\\end{proof}\n\n\\subsection{Vector field}\n\\label{sec:vector-field-Q}\n\n  It is possible to construct a vector field adapted to the presence\n  of $Q$, even though it is not needed to prove\n  Proposition~\\ref{prop:wave-class}. Such a vector field will be useful\n  in Section~\\ref{sec:cv}, and since its construction is very much in\n  the continuity of Section~\\ref{sec:mehler}, we present it now. Set,\n  for a scalar function $f$, \n\\begin{equation*}\n  {\\mathcal A} f= i W(t) e^{i\\phi(t,y)}\\nabla \\(\n    e^{-i\\phi(t,y)}f\\)= W(t) \\(f\\nabla \\phi +i\\nabla f\\),\n\\end{equation*}\nwhere $W$ is a matrix and the phase $\\phi$ solves the eikonal equation\n\\begin{equation*}\n \\d_t \\phi +\\frac{1}{2}|\\nabla \\phi|^2 + \\frac{1}{2}\\< Q(t)y,y\\>=0. \n\\end{equation*}\nSince the underlying Hamiltonian is quadratic, $\\phi$ has the form\n\\begin{equation*}\n  \\phi(t,y) = \\frac{1}{2}\\<K(t)y,y\\>,\n\\end{equation*}\nwhere $K(t)$ is a symmetric matrix. For $ {\\mathcal A}$ to commute\nwith $i\\d_t -H_Q$, we come up with the conditions\n\\begin{equation*}\n   \\dot K + K^2 + Q=0,\\quad  \\dot W = W \\nabla^2\\phi= WK .\n\\end{equation*}\nWe see that we can take $K=M_1$ as in the proof of\nLemma~\\ref{lem:riccati}, and $ {\\mathcal A}$ will then satisfy the\nsame three properties as $J$, up to the fact that the commutation\nproperty now includes the quadratic potential. \n\\smallbreak\n\nSince the construction of this vector field\nboils down to solving a matricial Riccati equation with initial data\nprescribed at large time (see \\eqref{eq:riccati}), we naturally\nconstruct two vector fields $\\mathcal A_\\pm$, associated to $t\\to \\pm\n\\infty$. In view of Lemma~\\ref{lem:riccati}, $\\mathcal A_-$ is defined\non $(-\\infty,-T]$, while $\\mathcal A_+$ is defined\non $[T,\\infty)$, for a common $T\\gg 1$, with\n\\begin{equation*}\n  \\mathcal A_\\pm  = W_\\pm(t)\\(\\nabla \\phi_\\pm + i\\nabla\\), \\quad\n  \\phi_\\pm (t,y) = \\frac{1}{2}\\<K_\\pm (t)y,y\\>,\n\\end{equation*}\nwhere $K_\\pm$ and $W_\\pm$ satisfy\n\\begin{equation*}\n  \\dot K_\\pm +K_\\pm^2+Q=0,\\quad \\dot W_\\pm =W_\\pm K_\\pm,\n\\end{equation*}\nso that Lemma~\\ref{lem:riccati} also yields\n\\begin{equation}\\label{eq:vector-asym}\n  K_\\pm(t)\\sim  \\frac{1}{t}{\\rm I}_d,\\quad  W_\\pm (t)\\sim t {\\rm\n    I}_d\\quad \\text{as }t\\to \\pm \\infty.  \n\\end{equation} \nWe construct commuting vector fields for large time only, essentially\nbecause on finite time intervals, the absence of commutation is not a\nproblem, so we can use $\\nabla$, $y$ or $J$. \n\n\n\n\\subsection{Asymptotic completeness}\n\\label{sec:ac-class}\n\nIn this section we prove:\n\\begin{proposition}\\label{prop:AC-class}\n  Let $d\\ge 1$, $\\frac{2}{d}\\le \\si<\\frac{2}{(d-2)_+}$, and $V$\n  satisfying Assumption~\\ref{hyp:V} for some $\\mu>1$. For all $u_0\\in\n  \\Sigma$, there exists a unique $\\tilde u_+\\in \\Sigma$ such that the\n  solution $u\\in C(\\R;\\Sigma)$ to \\eqref{eq:u} with $u_{\\mid t=0}=u_0$\n  satisfies\n  \\begin{equation*}\n   \\sum_{\\Gamma\\in \\{{\\rm Id},\\nabla, J\\}} \\|\\Gamma(t) u(t)-\n   \\Gamma(t)U_Q(t,0)\\tilde u_+\\|_{L^2} \\Tend t {+\\infty} 0. \n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  In the case $Q=0$, such a result is a rather direct consequence of\n  the \\emph{pseudo-conformal conservation law}, established in\n  \\cite{GV79Scatt}. Recalling that $J(t)=y+it\\nabla$, this law reads\n\\begin{equation*}\n  \\frac{d}{dt}\\(\\frac{1}{2}\\|J(t)u\\|_{L^2}^2\n  +\\frac{t^2}{\\si+1}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\)\n  =\\frac{t}{\\si+1}(2-d\\si)\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}. \n\\end{equation*}\nA way to derive this relation is to apply $J$ to \\eqref{eq:u}. The operator $J$\ncommutes with the linear part ($Q=0$), and the standard $L^2$\nestimate, which consists in multiplying the outcome by\n$\\overline{Ju}$, integrating in space, and taking the imaginary part, yields:\n\\begin{equation*}\n\\frac{1}{2} \\frac{d}{dt}\\|J(t)u\\|_{L^2}^2  = \\IM \\int \\overline {Ju}J\\(|u|^{2\\si}u\\).\n\\end{equation*}\nSince we have $J= i t  e^{i\\frac{|y|^2}{2t}} \\nabla\\( \\cdot\ne^{-i\\frac{|y|^2}{2t}}\\)$, \n\\begin{equation*}\n  J\\(|u|^{2\\si}u\\) = (\\si+1)|u|^{2\\si}Ju + \\si u^{\\si+1}\\bar\n  u^{\\si-1}\\overline{Ju}.\n\\end{equation*}\nThe first term is real, and the rest of the computation consists in\nexpanding the remaining term. \n\\smallbreak\n\nIn the case where $Q\\not =0$, we resume the above approach: the new\ncontribution is due to the fact that $J$ does not commute with the\nexternal potential, so we find:\n\\begin{align*}\n  \\frac{1}{2} \\frac{d}{dt}\\|J(t)u\\|_{L^2}^2 & =\\text{like before} + \\RE \\int\n  t Q(t)xu\\cdot \\overline {Ju}\\\\\n&=\\text{like before} + \n  t\\RE\\int_{\\R^d} \\<Q(t) J(t)u,J(t)u\\> +t^2 \\IM \\int_{\\R^d} \\<Q(t)\\nabla u,Ju\\>.\n\\end{align*}\nOn the other hand, we still have\n\\begin{align*}\n  \\frac{d}{dt}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}& =2 (\\si+1)\\int\n  |u|^{2\\si}\\RE \\(\\bar u\\d_tu\\) = 2 (\\si+1)\\int\n  |u|^{2\\si}\\RE \\(\\bar u \\times\\frac{i}{2}\\Delta u\\) ,\n\\end{align*}\nand so,\n\\begin{align*}\n  \\frac{d}{dt}\\(\\frac{1}{2}\\|J(t)u\\|_{L^2}^2\n  +\\frac{t^2}{\\si+1}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\)\n  &=\\frac{t}{\\si+1}(2-d\\si)\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\\\\n+ \n t\\RE\\int_{\\R^d} \\<Q(t) J(t)u,J(t)u\\>& +t^2 \\IM \\int_{\\R^d} \\<Q(t)\\nabla u,Ju\\> . \n\\end{align*}\nThus for $t\\ge 0$ and $\\si\\ge\\frac{2}{d}$, \\eqref{eq:decayQ} implies\n\\begin{equation*}\n  \\frac{d}{dt}\\(\\frac{1}{2}\\|J(t)u\\|_{L^2}^2\n  +\\frac{t^2}{\\si+1}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\)\n  \\lesssim  \n  \\<t\\>^{-\\mu-1}\\|J(t)u\\|_{L^2}^2 +\\<t\\>^{-\\mu}\\| \\nabla u\\|_{L^2}  \\|Ju\\|_{L^2}. \n\\end{equation*}\nEven though there is no conservation of the energy for \\eqref{eq:u}\nsince the potential depends on time, we know from \\cite{Ha13} that $u\\in\nL^\\infty(\\R;H^1(\\R^d))$. As a matter of fact, the proof given in\n\\cite[Section~4]{Ha13} concerns the case $\\si=1$ in $d=2$ or $3$, but\nthe argument, based on energy estimates,  remains valid for $d\\ge 1$,\n$\\si<\\frac{2}{(d-2)_+}$, since we then know that $u\\in\nC(\\R;\\Sigma)$. Since $\\mu>1$, we infer\n\\begin{equation}\\label{eq:Jborne}\n  Ju\\in L^\\infty(\\R_+;L^2). \n\\end{equation}\nWriting Duhamel's formula for \\eqref{eq:u} with initial datum $u_0$,\nin terms of $U_Q$, we have\n\\begin{equation*}\n  u(t) = U_Q(t,0)u_0-i\\int_0^t U_Q(t,s)\\(|u|^{2\\si}u(s)\\)ds.\n\\end{equation*}\nResuming the computations presented in the proof of\nProposition~\\ref{prop:wave-class}, \\eqref{eq:Jborne} and (weighted)\nGagliardo-Nirenberg inequalities make it possible to prove that\n\\begin{equation*}\n  Bu \\in L^{q_1}(\\R_+;L^{r_1}),\\ \\forall (q_1,r_1)\\text{ admissible},\n  \\ \\forall B\\in \\{{\\rm Id},\\nabla, J\\}. \n\\end{equation*}\nDuhamel's formula then yields, for $0<t_1<t_2$,\n\\begin{equation*}\n  U_Q(0,t_2)u(t_2)-U_Q(0,t_1)u(t_1) =\n  -i\\int_{t_1}^{t_2}U_Q(0,s)\\(|u|^{2\\si}u(s)\\)ds. \n\\end{equation*}\nFrom Strichartz estimates,\n\\begin{equation*}\n  \\| U_Q(0,t_2)u(t_2)-U_Q(0,t_1)u(t_1) \\|_{L^2}\\lesssim \\left\\|\n    |u|^{2\\si}u\\right\\|_{L^{q'}([t_1,t_2]:L^{r'})}, \n\\end{equation*}\nand the right hand side goes to zero as $t_1,t_2\\to\n+\\infty$. Therefore, there exists (a unique) $\\tilde u_+\\in L^2$ such that\n\\begin{equation*}\n   \\| U_Q(0,t)u(t)-\\tilde u_+ \\|_{L^2}\\Tend t {+\\infty} 0 ,\n\\end{equation*}\nand we have\n\\begin{equation*}\n  u(t) = U_Q(t,0)\\tilde u_+ +i\\int_t^\\infty\n  U_Q(t,s)\\(|u|^{2\\si}u(s)\\)ds. \n\\end{equation*}\nUsing the same estimates as in the proof of\nProposition~\\ref{prop:wave-class}, we infer\n\\begin{align*}\n  \\|\\nabla u(t) - \\nabla  U_Q(t,0)\\tilde u_+\\|_{L^2} &+  \\|J(t) u(t) -\n  J(t)  U_Q(t,0)\\tilde u_+\\|_{L^2} \\\\\n& \\lesssim \\left\\| |u|^{2\\si}\\nabla\n  u\\right\\|_{L^{q'}(t,\\infty;L^{r'})} +  \\left\\| |u|^{2\\si} J\n  u\\right\\|_{L^{q'}(t,\\infty;L^{r'})}  \\\\\n&\\quad+\n  \\|\\<s\\>^{-\\mu-1}J(s)u\\|_{L^1(t,\\infty;L^2)} +  \\|\\<s\\>^{-\\mu}\\nabla u\\|_{L^1(t,\\infty;L^2)}. \n\\end{align*}\nThe right hand side goes to zero as $t\\to \\infty$, hence the\nproposition. \n\\end{proof}\n\n\\begin{remark}\n  As pointed out in the previous section, it would be possible to\n  prove the existence of wave operators by using an adapted vector\n  field $\\mathcal A$. On the other hand, if $Q(t)$ is not proportional\n  to the identity matrix, it seems that no (exploitable) analogue of\n  the pseudo-conformal conservation law is available in terms of\n  $\\mathcal A$ rather than in terms of $J$. \n\\end{remark}\n\\subsection{Conclusion}\n\\label{sec:concl-class}\n\nLike in the case of quantum scattering, we use a stronger version of\nthe linear scattering theory:\n\\begin{proposition}\\label{prop:Cook-class}\n  Let $d\\ge 1$, $V$ satisfying Assumption~\\ref{hyp:V} with $\\mu>1$. Then \nthe strong limits\n \\begin{equation*}\n  \\lim_{t\\to \\pm \\infty} U_Q(0,t)U(t) \\quad \\text{and}\\quad  \\lim_{t\\to\n    \\pm\\infty} U(-t) U_Q(t,0) \\quad \\text{and}\\quad  \n\\end{equation*}\nexist in $\\Sigma$. \n\\end{proposition}\n\\begin{proof}\n  For the first limit (existence of wave operators), again in view of\n  Cook's method, we prove that for all $\\varphi\\in \n  \\Sch(\\R^d)$, \n\\begin{equation*}\n    t\\mapsto \\left\\| U_Q(0,t) \\<Q(t)y,y\\>U(t)\\varphi\\right\\|_{\\Sigma}\\in\n    L^1(\\R). \n  \\end{equation*}\nFor the $L^2$ norm, we have, in view of \\eqref{eq:decayQ},\n\\begin{equation*}\n  \\left\\| U_Q(0,t) \\<Q(t)y,y\\>U(t)\\varphi\\right\\|_{L^2} \\lesssim\n  \\<t\\>^{-\\mu-2}\\sum_{j=1}^d\\| y_j^2 U(t)\\varphi\\|_{L^2}.\n\\end{equation*}\nWrite\n\\begin{equation*}\n  y_j^2 = (y_j+it\\d_j)^2 +t^2\\d_j^2 -2ity_j\\d_j =  (y_j+it\\d_j)^2\n  -t^2\\d_j^2 -2it(y_j+it\\d_j)\\d_j,\n\\end{equation*}\nto take advantage of the commutation\n\\begin{equation*}\n  (y_j+it\\d_j)U(t) = U(t)y_j,\n\\end{equation*}\nand infer\n\\begin{equation*}\n  \\left\\| U_Q(0,t) \\<Q(t)y,y\\>U(t)\\varphi\\right\\|_{L^2} \\lesssim\n  \\<t\\>^{-\\mu-2}\\(\\||y|^2\\varphi\\|_{L^2} +t^2\\|\\Delta \\varphi\\|_{L^2}\n  \\)\\lesssim \\<t\\>^{-\\mu}.\n\\end{equation*}\nThe right hand side is integrable since $\\mu>1$, so the strong limits\n\\begin{equation*}\n   \\lim_{t\\to \\pm\\infty} U_Q(0,t)U(t)\n\\end{equation*}\nexist in $L^2$. \nTo infer that these strong limits actually exist in $\\Sigma$, we\nsimply invoke \\eqref{eq:Qhomo} in the case $(q_1,r_1)=(\\infty,2)$, so\nthe above computation are easily adapted. \n\\smallbreak\n\nFor asymptotic completeness, we can adopt the same strategy. Indeed,\nit suffices to prove that  for all $\\varphi\\in \n  \\Sch(\\R^d)$, \n\\begin{equation*}\n    t\\mapsto \\left\\| U(-t) \\<Q(t)y,y\\>U_Q(t,0)\\varphi\\right\\|_{\\Sigma}\\in\n    L^1(\\R). \n  \\end{equation*}\nFor the $L^2$ norm, we have\n\\begin{align*}\n  \\left\\| U(-t) \\<Q(t)y,y\\>U_Q(t,0)\\varphi\\right\\|_{L^2}&= \\left\\|\n   \\<Q(t)y,y\\>U_Q(t,0)\\varphi\\right\\|_{L^2}\\\\\n& \\lesssim\n \\<t\\>^{-\\mu-2}\\sum_{j=1}^d  \\left\\|\n   y_j^2U_Q(t,0)\\varphi\\right\\|_{L^2}.\n\\end{align*}\nWe first proceed like above, and write\n\\begin{equation*}\n  y_j^2 =  (y_j+it\\d_j)^2\n  -t^2\\d_j^2 -2it(y_j+it\\d_j)\\d_j.\n\\end{equation*}\nThe operator $J$ does not commute with $U_Q$, but this lack of\ncommutation is harmless for our present goal, from\n\\eqref{eq:Qhomo}. By considering the system satisfied by\n$$(y_j+it\\d_j)^2U_Q(t,0)\\varphi, \\d_j^2 U_Q(t,0)\\varphi,\n\\d_j(y_j+it\\d_j)U_Q(t,0)\\varphi,$$ \nwe obtain \n\\begin{align*}\n \\sum_{j=1}^d&\\( \\| (y_j+it\\d_j)^2U_Q(t,0)\\varphi\\|_{L^2} + \\|\\d_j^2\n U_Q(t,0)\\varphi\\|_{L^2} +\n \\|\\d_j(y_j+it\\d_j)U_Q(t,0)\\varphi\\|_{L^2}\\)\\\\\n&\\le C \\|\\varphi\\|_{\\Sigma^2},\n\\end{align*}\nwhere $\\Sigma^k$ is the space of $H^k$ functions with $k$ momenta in\n$L^2$, and $C$ does not depend on time. Finally, we also have a\nsimilar estimate by considering one more derivative or momentum. The\nkey remark in the computation is that the external\npotential $\\<Q(t)y,y\\>$ is exactly quadratic in space, and so\ndifferentiating it three times with any space variables yields zero. \n\\end{proof}\n\n\n\n\\section{Proof of Theorem~\\ref{theo:cv}}\n\\label{sec:cv}\n\nThe main result of this section is:\n\\begin{theorem}\\label{theo:cv-unif}\n  Let $d=3$, $\\si=1$, $V$ as in Theorem~\\ref{theo:scatt-quant}, and\n  $u_-\\in \\Sigma^7$. Suppose that Assumption~\\ref{hyp:flot} \n  is satisfied. Let $\\psi^\\eps$ be given by\n  Theorem~\\ref{theo:scatt-quant}, $u$ be given by\n  Theorem~\\ref{theo:scatt-class}, $\\varphi^\\eps$ defined by\n  \\eqref{eq:phi}.  We have\n  the uniform error \n  estimate:\n  \\begin{equation*}\n   \\sup_{t\\in \\R}\\|\\psi^\\eps(t)-\\varphi^\\eps(t)\\|_{L^2(\\R^3)}  =\n    \\O\\(\\sqrt\\eps\\). \n  \\end{equation*}\n\\end{theorem}\nTheorem~\\ref{theo:cv} is a direct consequence of the above\nresult, whose proof is the core of\nSection~\\ref{sec:cv}. From now on, we assume $d=3$ and $\\si=1$. \n\\subsection{Extra properties for the approximate solution}\n\\label{sec:extra-u}\n\nFurther regularity and localization properties on $u$ will be\nneeded. \n\\begin{proposition}\\label{prop:extra-u}\n  Let $\\si=1$, $1\\le d\\le 3$, $k\\ge 2$ and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>1$. If $u_-\\in \\Sigma^k$, then\n  the solution $u\\in C(\\R;\\Sigma)$ provided by Theorem~\\ref{theo:scatt-class}\n  satisfies $u\\in C(\\R;\\Sigma^k)$.  The momenta\n  of $u$ satisfy\n  \\begin{equation*}\n    \\lVert \\lvert y\\rvert^\\ell u(t,y)\\|_{L^2(\\R^d)}\\le C_\\ell\n    \\<t\\>^\\ell,\\quad 0\\le \\ell\\le k,\n  \\end{equation*}\nwhere $C_\\ell$ is independent of $t\\in \\R$.\n \\end{proposition}\n\\begin{proof}\n  We know from the proof of Theorem~\\ref{theo:scatt-class} that since\n  $u_-\\in \\Sigma$,\n\\begin{equation*}\n  u,\\nabla u, Ju \\in L^\\infty(\\R;L^2(\\R^d)).\n  \\end{equation*}\nThe natural approach is  then to proceed by induction on $k$,\nto prove that \n\\begin{align*}\n  \\nabla^k u,J^k u\\in L^\\infty(\\R;L^2(\\R^d)). \n\\end{align*}\n We have, as we have seen in the proof of\nProposition~\\ref{prop:wave-class},\n\\begin{align*}\n   i\\d_t \\nabla u &= H_Q \\nabla u + Q(t)y u +\\nabla\n                    \\(|u|^2 u\\)\\\\\n&+ H_Q \\nabla u + Q(t)J(t)u -it Q(t)\\nabla u +\\nabla\n                    \\(|u|^2 u\\),\\\\\ni\\d_t Ju  & = H_Q Ju   +it Q(t)y u +J\n                    \\(|u|^2 u\\)\\\\\n& = H_Q J u + itQ(t)J(t)u +t^2Q(t)\\nabla u +J\n                    \\(|u|^2 u\\).\n\\end{align*}\nApplying the operators $\\nabla$ and $J$ again, we find\n\\begin{align*}\n   i\\d_t \\nabla^2 u &= H_Q \\nabla^2 u + 2Q(t)y \\nabla u +Q(t) u +\\nabla^2\n                    \\(|u|^2 u\\)\\\\\n&+ H_Q \\nabla u + 2Q(t)J(t)\\nabla u -2it Q(t)\\nabla^2 u+Q(t)u +\\nabla^2\n                    \\(|u|^2 u\\),\\\\\ni\\d_t J^2u  & = H_Q J^2u   -2t^2 Q(t)y \\nabla u -t^2Q(t)u+J^2\n                    \\(|u|^2 u\\)\\\\\n& = H_Q J^2 u - 2t^2Q(t)J\\nabla u +2it^3Q(t) J^2 u +itQ(t)u+J^2\n                    \\(|u|^2 u\\).\n\\end{align*}\nIn view of \\eqref{eq:decayQ}, we see that $t\\mapsto t^3 Q(t)$ need not be\nintegrable (unless we make stronger and stronger assumptions of $\\mu$,\nas $k$ increases), so the commutator seems to be fatal to this approach. To\novercome this issue, we use the vector field mentioned in\nSection~\\ref{sec:vector-field-Q}. \nFor bounded time $t\\in\n[-T,T]$, the above mentioned lack of commutation is not a problem, and\nwe can use the operator $J$, which is defined for all time. \nWe note that either of the operators $\\mathcal A_\\pm$ or \n$J$ satisfies more generally the pointwise identity\n\\begin{equation*}\n  B\\(u_1\\overline u_2 u_3\\) =\\( B u_1\\) \\overline u_2 u_3 +\n  u_1\\(\\overline{B u_2}\\) u_3 + u_1\\overline u_2\\( Bu_3\\),\n\\end{equation*}\nfor all differentiable functions $u_1,u_2,u_3$. \n\nNow we have all the tools to proceed by induction, and mimic the\nproof from \\cite[Appendix]{Ca11}. The main idea is that the proof is\nsimilar to the propagation of higher regularity for energy-subcritical\nproblems, with the difference that large time is handled thanks to\nvector fields. We leave out the details, which are not difficult but\nrather cumbersome: considering\n\\begin{equation*}\n  B(t) =\n\\left\\{\n  \\begin{aligned}\n    \\mathcal A_-(t)&\\text{ for }t\\le -T,\\\\\nJ(t)&\\text{ for }t\\in [-T,T],\\\\\n \\mathcal A_+(t)&\\text{ for }t\\ge T,\n  \\end{aligned}\n\\right.\n\\end{equation*}\n we can then prove that \n \\begin{equation*}\n   \\nabla^k u,B^k u\\in L^\\infty(\\R;L^2(\\R^d)).  \n \\end{equation*}\nBack to the definition of $\\mathcal A_\\pm$, \n\\begin{equation*}\n  \\mathcal A_\\pm (t) = W_\\pm (t)K_\\pm (t)y +iW_\\pm (t)\\nabla,\n\\end{equation*}\n\\eqref{eq:vector-asym}\nthen yields the result. \n\\end{proof}\n\n\\subsection{Strichartz estimates}\n\\label{sec:strichartz-raff}\n\nIntroduce the following notations, taking the dependence upon $\\eps$\ninto account:\n\\begin{equation*}\n  H^\\eps=-\\frac{\\eps^2}{2}\\Delta+V(x),\\quad U_V^\\eps(t) =\n  e^{-i\\frac{t}{\\eps} H^\\eps}. \n\\end{equation*}\nSince we now work only in space dimension $d=3$, we can use the result\nfrom \\cite{Go06}. Resuming the proof from \\cite{Go06} (a mere scaling\nargument is not sufficient), we have, along with the preliminary\nanalysis from Section~\\ref{sec:spectral}, the global dispersive estimate\n\\begin{equation}\n  \\label{eq:disp-semi-glob}\n  \\|U^\\eps_V(t)\\|_{L^1(\\R^3)\\to L^\\infty(\\R^3)}\\lesssim\n  \\frac{1}{(\\eps|t|)^{3\/2}},\\quad t\\not =0.\n\\end{equation}\nFor $|t|\\le \\delta$, $\\delta>0$ independent of $\\eps$, the above\nrelation stems initially from \\cite{Fujiwara}. As a consequence, we\ncan measure the dependence upon $\\eps$ in Strichartz estimates. We\nrecall the definition of admissible pairs related to Sobolev\nregularity.\n\\begin{definition}\n  Let $d=3$ and $s\\in \\R$. A pair $(q,r)$ is called $\\dot H^s$-admissible if \n  \\begin{equation*}\n    \\frac{2}{q}+\\frac{3}{r} = \\frac{3}{2}-s. \n  \\end{equation*}\n\\end{definition}\nFor $t_0\\in \\R\\cup \\{-\\infty\\}$, we denote by\n\\begin{equation*}\n  R^\\eps_{t_0}(F)(t) = \\int_{t_0}^t U_V^\\eps(t-s)F(s)ds\n\\end{equation*}\nthe retarded term related to Duhamel's formula. Since the dispersive\nestimate \\eqref{eq:disp-semi-glob} is the same as the one for\n$e^{i\\eps t\\Delta}$, we get the same scaled Strichartz estimates as\nfor this operator, which can in turn be obtained by scaling\narguments from the case $\\eps=1$. \n\\begin{lemma}[Scaled $L^2$-Strichartz estimates]\\label{lem:stri-eps}\n  Let $t_0\\in \\R\\cup\\{-\\infty\\}$, and let $(q_1,r_1)$ and $(q_2,r_2)$\n  be $L^2$-admissible pairs, $2\\le  r_j\\le 6$. We have\n  \\begin{equation*}\n   \\eps^{\\frac{1}{q_1}} \\|U_V^\\eps(\\cdot) f\\|_{L^{q_1}(\\R;L^{r_1}(\\R^3))}\\lesssim\n    \\|f\\|_{L^2(\\R^3)}, \n  \\end{equation*}\n  \\begin{equation*}\n  \\eps^{\\frac{1}{q_1}+\\frac{1}{q_2}}\n  \\|R^\\eps_{t_0}(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^3))}\\le C_{q_1,q_2} \\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^3))},\n  \\end{equation*}\nwhere $C_{q_1,q_2}$ is independent of $\\eps$, $t_0$, and of $I$ such that\n$t_0\\in \\bar I$. \n\\end{lemma}\nWe will also use Strichartz estimates for non-admissible pairs, as\nestablished in \\cite{Kat94} (see also \\cite{CW92,FoschiStri}).\n\\begin{lemma}[Scaled inhomogeneous Strichartz\n  estimates]\\label{lem:stri-inhom-eps} \n   Let $t_0\\in \\R\\cup\\{-\\infty\\}$, and let $(q_1,r_1)$ be an $\\dot\n   H^{1\/2}$-admissible pair, and $(q_2,r_2)$\n  be an $\\dot H^{-1\/2}$-admissible pair, with \n  \\begin{equation*}\n    3\\le r_1,r_2<6.\n  \\end{equation*}\nWe have\n\\begin{equation*}\n  \\eps^{\\frac{1}{q_1}+\\frac{1}{q_2}}\n  \\|R^\\eps_{t_0}(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^3))}\\le C_{q_1,q_2} \\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^3))},\n  \\end{equation*}\nwhere $C_{q_1,q_2}$ is independent of $\\eps$, $t_0$, and of $I$ such that\n$t_0\\in \\bar I$. \n\\end{lemma}\n\\subsection{Preparing the proof}\n\\label{sec:preparing-proof}\n\n\n\n\n\n Subtracting the equations satisfied by\n$\\psi^\\eps$ and $\\varphi^\\eps$, respectively, we obtain as in\n\\cite{CaFe11}: $w^\\eps=\\psi^\\eps-\\varphi^\\eps$ satisfies\n\\begin{equation}\\label{eq:restecrit}\n  i\\eps\\d_t w^\\eps +\\frac{\\eps^2}{2}\\Delta w^\\eps =V w^\\eps -\\mathcal L^\\eps\n  + \n  \\eps^{5\/2}\\(|\\psi^\\eps|^{2}\\psi^\\eps\n  -|\\varphi^\\eps|^{2}\\varphi^\\eps\\),\n\\end{equation}\nalong with the initial condition\n\\begin{equation*}\n    e^{-i\\frac{\\eps\n      t}{2}\\Delta}w^\\eps_{\\mid t=-\\infty}=0,  \n\\end{equation*}\nwhere the source term is given by \n\\begin{equation*}\n  {\\mathcal L}^\\eps(t,x) = \\(V(x) - V\\(q(t)\\)\n  -\\sqrt\\eps \\<\\nabla V\\(q(t)\\),y\\>\n  -\\frac{\\eps}{2}\\<Q(t)y,y\\>\\)\\Big|_{y=\\frac{x-q(t)}{\\sqrt\\eps}}\n  \\varphi^\\eps(t,x). \n\\end{equation*}\nDuhamel's\nformula for $w^\\eps$ reads\n\\begin{align*}\n  w^\\eps(t) &= -i\\eps^{3\/2}\\int_{-\\infty}^t U^\\eps_V(t-s)\\(|\\psi^\\eps|^{2}\\psi^\\eps\n  -|\\varphi^\\eps|^{2}\\varphi^\\eps\\)(s)ds\\\\\n&\\quad  +i\\eps^{-1}\\int_{-\\infty}^t U^\\eps_V(t-s) \\mathcal L^\\eps(s)ds. \n\\end{align*}\nDenoting $L^a(]-\\infty,t];L^b(\\R^3))$ by $L^a_tL^b$, Strichartz\nestimates yield, for any $L^2$-admissible pair $(q_1,r_1)$,\n\\begin{equation}\\label{eq:stri-weps}\n  \\eps^{1\/q_1}\\|w^\\eps\\|_{L^{q_1}_t L^{r_1}} \\lesssim\n  \\eps^{3\/2-1\/q}\\left\\||\\psi^\\eps|^{2}\\psi^\\eps \n  -|\\varphi^\\eps|^{2}\\varphi^\\eps\\right\\|_{L^{q'}_tL^{r'}} +\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1_tL^2},\n\\end{equation}\nwhere $(q,r)$ is the admissible pair chosen in the proof of\nProposition~\\ref{prop:waveop-quant}, that is $r=2\\si+2$. Since we now\nhave  $d=3$ and $\\si=1$, this means:\n\\begin{equation*}\n  q=\\frac{8}{3},\\quad k=8,\n\\end{equation*}\nand \\eqref{eq:stri-weps} yields\n\\begin{equation}\\label{eq:w-presque}\n  \\eps^{1\/q_1}\\|w^\\eps\\|_{L^{q_1}_t L^{r_1}} \\lesssim\n  \\eps^{9\/8}\\( \\|w^\\eps\\|^2_{L^8_t L^4}+ \\|\\varphi^\\eps\\|^2_{L^8_t\n    L^4}\\)\\|w^\\eps\\|_{L^{8\/3}_tL^4}  +\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1_tL^2}.\n\\end{equation}\nThe strategy is then to first\nobtain an a priori estimate for $w^\\eps$ in $L^8_tL^4$, and then to\nuse it in the above estimate. In order to do so, we begin by\nestimating the source term $\\mathcal L^\\eps$, in the next subsection. \n\\subsection{Estimating the source term}\n\\label{sec:estim-source-term}\n\n\\begin{proposition}\\label{prop:est-source}\n  Let $d= 3$, $\\si=1$, $V$ satisfying Assumption~\\ref{hyp:V}\n  with $\\mu>2$, and $u_-\\in \\Sigma^k$ for some $k\\ge\n  7$. Suppose that Assumption~\\ref{hyp:flot} is satisfied.\nLet $u\\in C(\\R;\\Sigma^k)$ given by\n  Theorem~\\ref{theo:scatt-class} and \n  Proposition~\\ref{prop:extra-u}. The source term $\\mathcal L^\\eps$ satisfies\n  \\begin{equation*}\n   \\frac{1}{\\eps} \\|\\mathcal L^\\eps(t)\\|_{L^2(\\R^3)}\\lesssim \\frac{\\sqrt\n     \\eps}{\\<t\\>^{3\/2} }\\quad\\text{and}\\quad  \\frac{1}{\\eps}\n   \\|\\mathcal L^\\eps(t)\\|_{L^{3\/2}(\\R^3)}\\lesssim \\frac{\n     \\eps^{3\/4}}{\\<t\\>^{3\/2} },\n\\quad \\forall\n    t\\in \\R.\n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\nTo ease notation, we note that\n\\begin{equation*}\n \\frac{1}{\\eps} \\mathcal L^\\eps(t,x) = \\frac{1}{\\eps^{3\/4}} {\\mathcal\n    S}^\\eps(t,y)\\Big|_{y=\\frac{x-q(t)}{\\sqrt\\eps}}\n  e^{i(S(t)+ip(t)\\cdot (x-q(t)))\/\\eps}, \n\\end{equation*}\nwhere\n\\begin{equation*}\n  {\\mathcal S}^\\eps(t,y) = \\frac{1}{\\eps}\\( V\\(q(t)+y\\sqrt \\eps\\) -V\\(q(t)\\)\n  -\\sqrt\\eps \\<\\nabla V\\(q(t)\\),y\\> -\\frac{\\eps}{2}\\<Q(t)y,y\\>\\)u(t,y).\n\\end{equation*}\nIn particular,\n\\begin{equation*}\n  \\frac{1}{\\eps}\\| \\mathcal L^\\eps(t)\\|_{L^2(\\R^3)} = \\|{\\mathcal\n    S}^\\eps(t)\\|_{L^2(\\R^3)} ,\\quad  \\frac{1}{\\eps}\\| \\mathcal\n  L^\\eps(t)\\|_{L^{3\/2}(\\R^3)} = \\eps^{1\/4}\\|{\\mathcal \n    S}^\\eps(t)\\|_{L^{3\/2}(\\R^3)}. \n\\end{equation*}\n  Taylor's formula and Assumption~\\ref{hyp:V} yield the pointwise estimate\n  \\begin{equation*}\n    | {\\mathcal S}^\\eps(t,y) | \\lesssim \\sqrt\\eps |y|^3 \\int_0^1\n    \\frac{1}{\\<q(t)+\\theta y\\sqrt \\eps\\>^{\\mu+3}}d\\theta |u(t,y)|. \n  \\end{equation*}\nTo simplify notations, we consider only positive times. Recall that\nfrom Assumption~\\ref{hyp:flot}, $p^+\\not =0$. Introduce, for\n$0<\\eta< |p^+|\/2$,\n\\begin{equation*}\n  \\Omega  = \\left\\{y\\in \\R^3,\\quad |y|\\ge \\eta\\frac{t}{\\sqrt\\eps}\\right\\}.\n\\end{equation*}\nSince $q(t)\\sim p^+ t$ as\n$t\\to \\infty$, on the complement of $\\Omega$, we can use the decay of $V$,\n\\eqref{eq:short}, to infer the pointwise estimate\n\\begin{equation}\\label{eq:Spoint}\n   | {\\mathcal S}^\\eps(t,y) | \\lesssim \\sqrt\\eps |y|^3\n    \\frac{1}{\\<t\\>^{\\mu+3}} |u(t,y)| \\quad \\text{on }\\Omega^c.\n\\end{equation}\nTaking the $L^2$-norm, we have\n\\begin{equation*}\n  \\|\\mathcal S^\\eps(t)\\|_{L^2(\\Omega^c)}\\le \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu+3}}\\||y|^3u(t,y)\\|_{L^2(\\R^3)}\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu}},\n\\end{equation*}\nwhere we have used Proposition~\\ref{prop:extra-u}. On $\\Omega$\nhowever, the argument of the potential in Taylor's formula is not\nnecessarily going to infinity, so the decay of the potential is\napparently useless. Back to the definition of $\\mathcal L^\\eps$, that is leaving\nout Taylor's formula, we see that all the terms but the first one can\nbe easily estimated on $\\Omega$. Indeed, the definition of $\\Omega$ implies\n\\begin{equation*}\n  |V(q(t))u(t,y)| \\lesssim \\frac{1}{\\<t\\>^\\mu}|u(t,y)|\\lesssim\n  \\frac{1}{\\<t\\>^\\mu} \\left| \\frac{y\\sqrt \\eps}{t}\\right|^k |u(t,y)|,\n\\end{equation*}\nwhere $k$ will be chosen shortly. Taking the $L^2$ norm, we find\n\\begin{equation*}\n  \\frac{1}{\\eps}\\|V(q(t))u(t)\\|_{L^2(\\Omega)} \\lesssim\n  \\frac{\\eps^{k\/2-1}}{\\<t\\>^{\\mu+k}}\\||y|^k u(t,y)\\|_{L^2(\\R^3)}\n  \\lesssim \\frac{\\eps^{k\/2-1}}{\\<t\\>^{\\mu}},\n\\end{equation*}\nwhere we have used Proposition~\\ref{prop:extra-u} again. Choosing\n$k=3$ yields the expected estimate. The last two terms in $\\mathcal\nL^\\eps$ can be estimated accordingly. For the first term in $\\mathcal\nL^\\eps$ however, we face the same problem as above: the argument of\n$V$ has to be considered as bounded. A heuristic argument goes as\nfollows. In view of Theorem~\\ref{theo:scatt-class},\n\\begin{equation*}\n  u(t,y) \\Eq t \\infty e^{i\\frac{t}{2}\\Delta}u_+ \\Eq t \\infty\n  \\frac{1}{t^{3\/2}}\\widehat u_+\\(\\frac{y}{t}\\)e^{i|y|^2\/(2t)},\n\\end{equation*}\nwhere the last behavior stems from standard analysis of the\nSchr\\\"odinger group (see e.g. \\cite{Rauch91}). In view of\nthe definition of $\\Omega$, we have, formally for $y\\in \\Omega$,  \n\\begin{equation*}\n  |u(t,y)|\\lesssim\n  \\frac{1}{t^{3\/2}}\\sup_{ |z|\\ge \\eta}\\left |\\widehat\n    u_+\\(\\frac{z}{\\sqrt\\eps}\\)\\right|. \n\\end{equation*}\nThen the idea is to keep the linear dispersion measured by the factor\n$t^{-3\/2}$ (which is integrable since $d=3$), and use decay properties\nfor $\\widehat u_+$ to gain powers of $\\eps$. To make this argument\nrigorous, we keep the idea \nthat $u$ must be assessed in $L^\\infty$ rather than in $L^2$, and write\n\\begin{equation*}\n  \\frac{1}{\\eps}\\|V\\(q(t)+y\\sqrt \\eps\\)u(t,y)\\|_{L^2(\\Omega)} \\le\n  \\frac{1}{\\eps}\\|u(t)\\|_{L^\\infty(\\Omega)} \\|V\\(q(t)+y\\sqrt \\eps\\)\\|_{L^2(\\Omega)} .\n\\end{equation*}\nFor the last factor, we have\n\\begin{equation*}\n  \\|V\\(q(t)+y\\sqrt\n  \\eps\\)\\|_{L^2(\\Omega)}\\le \\eps^{-3\/4}\\|V\\|_{L^2(\\R^3)},\n\\end{equation*}\nwhere the last norm is finite since $\\mu>2$. For the $L^\\infty$ norm of\n$u$, we use Gagliardo-Nirenberg inequality and the previous\nvector-fields. To take advantage of the localization in space,\nintroduce a non-negative cut-off function $\\chi\\in C^\\infty(\\R^3)$, such that:\n\\begin{equation*}\n  \\chi(z)=\\left\\{\n    \\begin{aligned}\n      1& \\text{ if }|z|\\ge \\eta,\\\\\n0 & \\text{ if }|z|\\le\\frac{\\eta}{2}.\n    \\end{aligned}\n\\right.\n\\end{equation*}\nIn view of the definition of $\\Omega$, \n\\begin{equation*}\n   \\|u(t)\\|_{L^\\infty(\\Omega)}  \\le \\left\\|\n     \\chi\\(\\frac{y\\sqrt\\eps}{t}\\)u(t,y)\\right\\|_{L^\\infty(\\R^3)}. \n\\end{equation*}\nNow with $B$ as defined in the proof of\nProposition~\\ref{prop:extra-u}, Gagliardo-Nirenberg inequality yields,\nfor any smooth function $f$\n (recall that $y\\in \\R^3$),\n \\begin{equation*}\n   \\|f\\|_{L^\\infty(\\R^3)}\\lesssim\n   \\frac{1}{t^{3\/2}}\\|f\\|_{L^2(\\R^3)}^{1\/4}\\|B^2(t)f\\|_{L^2(\\R^3)}^{3\/4}.\n \\end{equation*}\nWe use this inequality with\n\\begin{equation*}\n  f(t,y) = \\chi\\(\\frac{y\\sqrt\\eps}{t}\\)u(t,y),\n\\end{equation*}\nand note that\n\\begin{equation*}\n  B(t)f (t,y)= \\chi\\(\\frac{y\\sqrt\\eps}{t}\\)B(t)u(t,y) + i\\frac{\\sqrt\n    \\eps}{t}W(t) \\nabla \\chi \\(\\frac{y\\sqrt\\eps}{t}\\) \\times u(t,y),\n\\end{equation*}\nwhere $W(t)$ stands for $W_\\pm$ or $t$. Recall that $t\\mapsto W(t)\/t$\nis bounded, so the last term is actually ``nice''. \nProceeding in the same way as above, we obtain\n\\begin{equation*}\n  \\|u(t)\\|_{L^2(\\Omega)}\\lesssim\n  \\left\\| \\left| \\frac{y\\sqrt \\eps}{t}\\right|^k\n    u(t,y)\\right\\|_{L^2(\\Omega)}\\lesssim \\eps^{k\/2},\n\\end{equation*}\nprovided that $u_-\\in \\Sigma^k$. Similarly,\n\\begin{equation*}\n  \\|B^2(t)u\\|_{L^2(\\Omega)} \\lesssim \\eps^{k\/2-1},\n\\end{equation*}\nand so \n\\begin{equation*}\n   \\frac{1}{\\eps}\\|V\\(q(t)+y\\sqrt \\eps\\)u(t,y)\\|_{L^2(\\Omega)}\\lesssim \n   \\frac{1}{t^{3\/2}}\\eps^{-7\/4 +k\/8+3(k\/2-1)\/4}= \\frac{\\eps^{k\/2-5\/2}}{t^{3\/2}}.\n\\end{equation*}\nTherefore, the $L^2$ estimate follows as soon as $k \\ge 6$. For the\n$L^{3\/2}$-estimate, we resume the same computations, and use the extra\nestimate: for all $s>1\/2$, \n\\begin{equation}\\label{eq:localizing}\n  \\|f\\|_{L^{3\/2}(\\R^3)}\\lesssim \\|f\\|_{L^2(\\R^3)}^{1-1\/2s}\\||x|^sf\\|_{L^2(\\R^3)}^{1\/2s}.\n\\end{equation}\nThis estimate can easily be proven by writing\n\\begin{equation*}\n   \\|f\\|_{L^{3\/2}(\\R^3)} \\le \\|f\\|_{L^{3\/2}(|y|<R)} + \\left\\|\n     \\frac{1}{|x|^s} |x|^s f\\right\\|_{L^{3\/2}(|x|>R)},\n\\end{equation*}\nso H\\\"older inequality yields, provided that $s>1\/2$ (so that\n$y\\mapsto |y|^{-s}\\in L^6(|y|>R)$)\n\\begin{equation*}\n   \\|f\\|_{L^{3\/2}(\\R^3)} \\le \\sqrt R \\|f\\|_{L^2} + \\frac{1}{R^{s-1\/2}}\\||x|^s f\\|_{L^2},\n\\end{equation*}\nand by optimizing in $R$. Now from \\eqref{eq:Spoint}, we have\n\\begin{align*}\n  \\|\\mathcal S^\\eps(t)\\|_{L^{3\/2}(\\Omega^c)}&\\le \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu+3}}\\||y|^3u(t,y)\\|_{L^{3\/2}(\\R^d)}\\\\\n&\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu+3}}\\||y|^3u(t,y)\\|_{L^2(\\R^d)}^{1\/2}\\||y|^4u(t,y)\\|_{L^2(\\R^d)}^{1\/2}\\\\\n&\n\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu-1\/2}}\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{3\/2}}\n\\end{align*}\nwhere we have used \\eqref{eq:localizing} with $s=1$, \nProposition~\\ref{prop:extra-u}, and the fact that $\\mu>2$. \n\nOn $\\Omega$, we can repeat the computations from the $L^2$-estimate\n(up to incorporating \\eqref{eq:localizing}): for the last term, we\nnote that\n\\begin{equation*}\n  \\frac{1}{\\eps}\\|V\\(q(t)+y\\sqrt \\eps\\)u(t,y)\\|_{L^{3\/2}(\\Omega)} \\le\n  \\frac{1}{\\eps}\\|u(t)\\|_{L^\\infty(\\Omega)} \\|V\\(q(t)+y\\sqrt \\eps\\)\\|_{L^{3\/2}(\\Omega)} ,\n\\end{equation*}\nand that\n\\begin{equation*}\n  \\|V\\(q(t)+y\\sqrt\n  \\eps\\)\\|_{L^{3\/2}(\\Omega)}\\le \\eps^{-1}\\|V\\|_{L^{3\/2}(\\R^3)},\n\\end{equation*}\nwhere the last norm is finite since $\\mu>2$. Up to taking $u$ in\n$\\Sigma^7$, we conclude\n\\begin{equation*}\n   \\|\\mathcal S^\\eps(t)\\|_{L^{3\/2}(\\R^3)}\\lesssim \\frac{\\sqrt\\eps}{\\<t\\>^{3\/2}},\n\\end{equation*}\nand the proposition follows. \n\\end{proof}\n\n\\subsection{A priori estimate for the error in the critical norm}\n\\label{sec:priori-estim-error}\n\nIn this subsection, we prove:\n\\begin{proposition}\\label{prop:w-crit}\n  Under the assumptions of Theorem~\\ref{theo:cv-unif}, the error\n  $w^\\eps=\\psi^\\eps-\\varphi^\\eps$  satisfies the a priori estimate,\n  for any $\\dot H^{1\/2}$-admissible pair \n  $(q,r)$, \n  \\begin{equation*}\n    \\eps^{\\frac{1}{q}}\\|w^\\eps\\|_{L^q(\\R;L^r(\\R^3))}\\lesssim \\eps^{1\/4}. \n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  The reason for considering $\\dot H^{1\/2}$-admissible pairs is that\n  the cubic three-dimensional Schr\\\"odinger equation is $\\dot\n  H^{1\/2}$-critical; see e.g. \\cite{CW90}. The proof of\n  Proposition~\\ref{prop:w-crit} is then very similar to the proof of\n  \\cite[Proposition~2.3]{HoRo08}. \n\nAn important tool is the known estimate for the approximate solution\n$\\varphi^\\eps$: we have, in view of the fact\nthat $u,Bu\\in L^\\infty L^2$,\n\\begin{equation}\\label{eq:est-a-priori-phi}\n  \\|\\varphi^\\eps(t)\\|_{L^r(\\R^3)}\\lesssim\n  \\(\\frac{1}{\\<t\\>\\sqrt\\eps}\\)^{3\\(\\frac{1}{2}-\\frac{1}{r}\\)},\\quad\n  2\\le r\\le 6.\n\\end{equation}\nNote that for an $\\dot H^{1\/2}$ admissible pair, we infer\n\\begin{equation*}\n  \\|\\varphi^\\eps(t)\\|_{L^q(\\R;L^r(\\R^3))}\\lesssim\n  \\eps^{-\\frac{3}{2}\\(\\frac{1}{2}-\\frac{1}{r}\\)} = \\eps^{-\\frac{1}{q}-\\frac{1}{4}},\n\\end{equation*}\nso Proposition~\\ref{prop:w-crit} shows a $\\sqrt\\eps$ gain\nfor $w^\\eps$ compared to $\\varphi^\\eps$, which is the order of\nmagnitude we eventually prove in $L^\\infty L^2$, and stated in\nTheorem~\\ref{theo:cv-unif}.  \nLet $0<\\eta\\ll 1$, and set\n\\begin{equation*}\n  \\|w^\\eps\\|_{\\mathcal N^\\eps(I)} :=\\sup_{{(q,r)\\ \\dot\n      H^{1\/2}-\\text{admissible}}\\atop 3\\le r\\le\n    6-\\eta}\\eps^{\\frac{1}{q}}\\|w^\\eps\\|_{L^q(I;L^r(\\R^3)}. \n\\end{equation*}\nDuhamel's formula for \\eqref{eq:restecrit} reads, given $w^\\eps_{\\mid\n  t=-\\infty}=0$,\n\\begin{equation*}\n  w^\\eps(t) =-i\\eps^{3\/2} \\int_{-\\infty}^t\n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\n+i\\eps^{-1} \\int_{-\\infty}^t  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds. \n\\end{equation*}\nSince we have the point-wise estimate\n\\begin{equation*}\n  \\left|\n    |\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\right|\\lesssim\n  \\(|w^\\eps|^2+|\\varphi^\\eps|^2\\)|w^\\eps|, \n\\end{equation*}\nLemma~\\ref{lem:stri-inhom-eps} yields, with\n$(q_2,r_2)=(\\frac{10}{7},5)$ for the first term of the right hand\nside, and with $(q_2,r_2)=(2,3)$ for the second term,\n\\begin{align*}\n  \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}&\\lesssim \\eps^{3\/2-7\/10}\\left\\|\n    \\(|w^\\eps|^2+|\\varphi^\\eps|^2\\)w^\\eps\\right\\|_{L^{10\/3}_tL^{5\/4}}\n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}}\\\\\n&\\lesssim \\eps^{4\/5}\\( \\|w^\\eps\\|_{L^{20}_tL^{10\/3}}^2 +\n\\|\\varphi^\\eps\\|_{L^{20}_tL^{10\/3}}^2 \\)\n\\|w^\\eps\\|_{L^{5}_tL^{5}}\n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}},\n\\end{align*}\nwhere we have used H\\\"older inequality. Note that the pairs\n$(20,\\frac{10}{3})$ and $(5,5)$ are $\\dot H^{1\/2}$-admissible. Denote\nby \n\\begin{equation*}\n  \\omega(t) =\\frac{1}{\\<t\\>^{3\/5}}. \n\\end{equation*}\nThis function obviously belongs to $L^{20}(\\R)$. \nThe estimate \\eqref{eq:est-a-priori-phi} and the definition of the\nnorm $\\mathcal N^\\eps$ yield\n\\begin{equation*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}\\lesssim \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}^3 + \n\\|\\omega\\|_{L^{20}(-\\infty,t)}^2  \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}\n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}}.\n\\end{equation*}\nTaking $t\\ll -1$, we infer\n\\begin{equation*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}\\lesssim \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}^3 \n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}}\\lesssim \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}^3  + \\eps^{1\/4},\n\\end{equation*}\nwhere we have use Proposition~\\ref{prop:est-source}. We can now use a\nstandard bootstrap argument, as recalled in Section~\\ref{sec:class}.\nWe infer that for $t_1\\ll -1$,\n\\begin{equation*}\n    \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t_1)}\\lesssim \\eps^{1\/4}.\n\\end{equation*}\nUsing Duhamel's formula again, we have\n\\begin{align*}\n    U_V^\\eps(t-t_1)w^\\eps(t_1) &=-i\\eps^{3\/2} \\int_{-\\infty}^{t_1}\n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\\\\\n&+i\\eps^{-1} \\int_{-\\infty}^{t_1}  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds,\n\\end{align*}\nso we infer\n\\begin{align*}\n    \\| U_V^\\eps(t-t_1)w^\\eps(t_1)\\|_{\\mathcal N^\\eps(\\R)}& \\lesssim  \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t_1)}^3 + \n\\|\\omega\\|_{L^{20}(-\\infty,t_1)}^2  \\|w^\\eps\\|_{\\mathcal\n                                                           N^\\eps(-\\infty,t_1)}\\\\\n& \\quad  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2((-\\infty,t_1]; L^{3\/2})}\\\\\n&\\le\nC_0\\eps^{1\/4}.\n\\end{align*}\nWe now rewrite Duhamel's formula with some initial time $t_j$:\n\\begin{align*}\n  w^\\eps(t) &= U_V^\\eps(t-t_j)w^\\eps(t_j)-i\\eps^{3\/2} \\int_{t_j}^t\n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\\\\\n&\\quad\n+i\\eps^{-1} \\int_{t_j}^t  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds. \n\\end{align*}\nFor $t\\ge t_j$ and $I=[t_j,t]$, the same estimates as above yield\n\\begin{align*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I)}&\\le  \\|U_V^\\eps(\\cdot-t_j)w^\\eps(t_j)\\|_{\\mathcal N^\\eps(I)}\n  + C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I)}^3 + \nC\\|\\omega\\|_{L^{20}(I)}^2  \\|w^\\eps\\|_{\\mathcal N^\\eps(I)}\\\\\n&  \\quad+ C\\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2(I; L^{3\/2})},\n\\end{align*}\nwhere the above constant $C$ is independent of $\\eps, t_j$ and $t$. We\nsplit $\\R_t$ into finitely many intervals\n\\begin{equation*}\n  \\R = (-\\infty,t_1]\\cup \\bigcup_{j=1}^N [t_j,t_{j+1}]\\cup\n[t_N,\\infty)=:\\bigcup_{j=0}^{N+1} I_j, \n\\end{equation*}\n on which \n\\begin{equation*}\n  C\\|\\omega\\|_{L^{20}(I_j)}^2 \\le\\frac{1}{2},\n\\end{equation*}\nso that we have\n\\begin{align*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}&\\le   2\\|U_V^\\eps(\\cdot-t_j)w^\\eps(t_j)\\|_{\\mathcal N^\\eps(I_j)}\n  +2 C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}^3  +\n  2 C\\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2(I_j; L^{3\/2})}\\\\\n&\\le  2\\|U_V^\\eps(\\cdot-t_j)w^\\eps(t_j)\\|_{\\mathcal N^\\eps(I_j)}\n  +2 C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}^3  + \\tilde C\n  \\eps^{1\/4}\\left\\|\\<t\\>^{-3\/2}\\right\\|_{L^2(I_j)}, \n\\end{align*}\nwhere we have used Proposition~\\ref{prop:est-source} again. Since we\nhave\n\\begin{equation*}\n   \\| U_V^\\eps(t-t_1)w^\\eps(t_1)\\|_{\\mathcal N^\\eps(\\R)}\\le C_0\\eps^{1\/4},\n\\end{equation*}\nthe bootstrap argument shows that at least for $\\eps\\le \\eps_1$\n($\\eps_1>0$),\n\\begin{equation*}\n    \\|w^\\eps\\|_{\\mathcal N^\\eps(I_1)}\\le 3\n    \\|U_V^\\eps(\\cdot-t_1)w^\\eps(t_1)\\|_{\\mathcal N^\\eps(I_1)} + \\frac{3}{2} \\tilde C\n  \\eps^{1\/4}\\left\\|\\<t\\>^{-3\/2}\\right\\|_{L^2(I_1)}.\n\\end{equation*}\nOn the other hand, Duhamel's formula implies\n\\begin{align*}\n  U_V^\\eps(t-t_{j+1})w^\\eps(t_{j+1}) &=\n                                       U_V^\\eps(t-t_j)w^\\eps(t_j)\n+i\\eps^{-1} \\int_{t_j}^{t_{j+1}}  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds\\\\\n&\\quad -i\\eps^{3\/2}\n                                       \\int_{t_j}^{t_{j+1}} \n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\n. \n\\end{align*}\nTherefore, we infer\n\\begin{align*}\n    \\| U_V^\\eps(t-t_{j+1})w^\\eps(t_{j+1})\\|_{\\mathcal N^\\eps(\\R)}&\\le\n    \\|U_V^\\eps(t-t_{j})w^\\eps(t_{j})\\|_{\\mathcal N^\\eps(\\R)}+ \n  + C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}^3 \\\\\n&  \\quad + \nC\\|\\omega\\|_{L^{20}(I_j)}^2  \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}\n+ C\\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2(I_j; L^{3\/2})}.\n\\end{align*}\nBy induction (carrying over finitely many steps), we conclude\n\\begin{equation*}\n   \\|U_V^\\eps(t-t_{j})w^\\eps(t_{j})\\|_{\\mathcal N^\\eps(\\R)}\n   =\\O\\(\\eps^{1\/4}\\),\\quad 0\\le j\\le N+1,\n\\end{equation*}\nand $\\|w^\\eps\\|_{\\mathcal N^\\eps(\\R)}=\\O\\(\\eps^{1\/4}\\)$ as announced. \n\\end{proof}\n\n\\subsection{End of the argument}\n\\label{sec:end-argument}\n\nResume the estimate \\eqref{eq:w-presque} with the $L^2$-admissible\npair $(q_1,r_1)= (\\frac{8}{3},4)$:\n\\begin{equation*}\n   \\eps^{3\/8}\\|w^\\eps\\|_{L^{8\/3}_t L^{4}} \\lesssim\n  \\eps^{3\/4}\\( \\|w^\\eps\\|^2_{L^8_t L^4}+ \\|\\varphi^\\eps\\|^2_{L^8_t\n    L^4}\\) \\eps^{3\/8}\\|w^\\eps\\|_{L^{8\/3}_tL^4}  +\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1_tL^2}.\n\\end{equation*}\nFrom Proposition~\\ref{prop:w-crit} (the pair $(8,4)$ is $\\dot H^{1\/2}$-admissible),\n\\begin{equation*}\n  \\|w^\\eps\\|_{L^8(\\R; L^4)}\\lesssim \\eps^{1\/8},\n\\end{equation*}\nand we have seen in the course of the proof that\n\\begin{equation*}\n  \\|\\varphi^\\eps\\|_{L^8(\\R; L^4)}\\lesssim \\eps^{-3\/8}.\n\\end{equation*}\nTherefore, we can split $\\R_t$ into finitely many intervals, in a way\nwhich is independent of $\\eps $, so that \n\\begin{equation*}\n   \\eps^{3\/4}\\( \\|w^\\eps\\|^2_{L^8(I; L^4)}+ \\|\\varphi^\\eps\\|^2_{L^8(I;\n    L^4)}\\)\\le \\eta \n\\end{equation*}\non each of these intervals, with $\\eta$ so small that we infer\n\\begin{equation*}\n   \\eps^{3\/8}\\|w^\\eps\\|_{L^{8\/3}(\\R; L^{4})} \\lesssim\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1(\\R;L^2)}\\lesssim \\sqrt\\eps,\n\\end{equation*}\nwhere we have used Proposition~\\ref{prop:est-source}. Plugging this\nestimate into  \\eqref{eq:w-presque} and now taking $(q_1,r_1)$,\nTheorem~\\ref{theo:cv-unif} follows.\n\n\n\n\n\n\\section{Superposition}\n\\label{sec:superp}\n\nIn this section, we sketch the proof of\nCorollary~\\ref{cor:decoupling}. This result heavily relies on the\n(finite time) superposition \n  principle established in \\cite{CaFe11}, in the case of two initial\n  coherent states with different centers in phase space. We present\n  the argument in the case of two initial wave packets, and explain\n  why it can be generalized to any finite number of initial coherent\n  states. \n\\smallbreak\n\nFollowing the proof of\n\\cite[Proposition~1.14]{CaFe11}, we introduce the approximate\nevolution of each individual initial wave packet:\n\\begin{equation*}\n   \\varphi_j^\\eps(t,x)=\\eps^{-3\/4} u_j\n\\left(t,\\frac{x-q_j(t)}{\\sqrt\\eps}\\right)e^{i\\left(S_j(t)+p_j(t)\\cdot\n    (x-q_j(t))\\right)\/\\eps},\n\\end{equation*}\nwhere $u_j$ solves \\eqref{eq:u} with initial datum $a_j$. In the proof\nof \\cite[Proposition~1.14]{CaFe11}, the main remark is that all that\nis needed is the control of a new source term, corresponding to the\ninteractions of the approximate solutions. Set \n\\begin{equation*}\n  w^\\eps = \\psi^\\eps -\\varphi_1^\\eps-\\varphi_2^\\eps.\n\\end{equation*}\nIt solves \n\\begin{equation*}\n   i\\eps\\d_t w^\\eps +\\frac{\\eps^2}{2}\\Delta w^\\eps = Vw^\\eps -\\mathcal\n  L^\\eps+\\mathcal N_I^\\eps+ \\mathcal N_s^\\eps\\quad ;\\quad w^\\eps_{\\mid t=0}=0,\n\\end{equation*}\nwhere the linear source term is the same as in Section~\\ref{sec:cv}\n(except than now we consider the sums of two such terms), $\\mathcal\nN_s^\\eps$ is the semilinear term\n\\begin{equation*}\n  \\mathcal N_s^\\eps =\\eps^{5\/2}\\( |w^\\eps+\n  \\varphi_1^\\eps+\\varphi_2^\\eps|^2 (w^\\eps+\n  \\varphi_1^\\eps+\\varphi_2^\\eps) - |\n  \\varphi_1^\\eps+\\varphi_2^\\eps|^2 (\n  \\varphi_1^\\eps+\\varphi_2^\\eps)\\),\n\\end{equation*}\nand $\\mathcal N_I^\\eps$ is precisely the new interaction term,\n\\begin{equation*}\n   \\mathcal N_I^\\eps =\\eps^{5\/2}\\( |\n  \\varphi_1^\\eps+\\varphi_2^\\eps|^2 (\n  \\varphi_1^\\eps+\\varphi_2^\\eps) - |\n  \\varphi_1^\\eps|^2 \\varphi_1^\\eps-|\n  \\varphi_2^\\eps|^2  \\varphi_2^\\eps\\).\n\\end{equation*}\nIn \\cite{CaFe11}, it is proven that if $(q_{01},p_{01})\\not\n=(q_{02},p_{02})$, then the possible interactions between\n$\\varphi_1^\\eps$ and $\\varphi_2^\\eps$ are negligible on every finite\ntime interval, in the sense that\n\\begin{equation*}\n  \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps\\|_{L^1(0,T;L^2)}\\le C(T,\\gamma) \\eps^\\gamma,\n\\end{equation*}\nfor every $\\gamma<1\/2$. We infer that\n$\\|w^\\eps\\|_{L^\\infty(0,T;L^2)}=\\O(\\eps^\\gamma)$ for every $T>0$. For\n$t\\ge T$, we have\n\\begin{align*}\n   \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps(t)\\|_{L^2}& \\lesssim\n  \\sum_{\\ell_1,\\ell_2\\ge 1,\\ \\ell_1+\\ell_2 =3}\\left\\|\n  u_1^{\\ell_1}\\(t,y-\\frac{q_1(t)-q_2(t)}{\\sqrt\\eps } \\)\n  u_2^{\\ell_2}(t,y)\\right\\|_{L^2}\\\\\n&\\lesssim\n  \\sum_{\\ell_1,\\ell_2\\ge 1,\\ \\ell_1+\\ell_2 =3}\\|\n  u_1(t)\\|_{L^\\infty}^{\\ell_1} \n\\| u_2(t)\\|_{L^\\infty}^{\\ell_2-1} \n\\|  u_2(t)\\|_{L^2}\\lesssim \\frac{1}{t^3}.\n\\end{align*}\nSimilarly, resuming the same estimates as in the proof of\nProposition~\\ref{prop:est-source}, \n\\begin{equation*}\n   \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps(t)\\|_{L^{3\/2}}\\lesssim\n   \\frac{\\eps^{1\/4}}{t^{5\/2}}. \n\\end{equation*}\nBy resuming the proof of Theorem~\\ref{theo:cv-unif} on the time\ninterval $[T,\\infty)$, we infer\n\\begin{equation*}\n  \\|w^\\eps\\|_{L^\\infty(0,\\infty;L^2)}\\le C(T,\\gamma)\\eps^\\gamma + \\frac{C}{T^2}.\n\\end{equation*}\nTherefore, \n\\begin{equation*}\n  \\limsup_{\\eps\\to 0}  \\|w^\\eps\\|_{L^\\infty(0,\\infty;L^2)}\\lesssim \\frac{1}{T^2},\n\\end{equation*}\nfor all $T>0$, hence the result by letting $T\\to \\infty$. \n\\smallbreak\n\nIn the case of more than two initial coherent states, the idea is that\nthe nonlinear interaction term, $\\mathcal N_I^\\eps$, always contains\nthe product of two approximate solutions corresponding to different\ntrajectories in phase space. This is enough for the proof of\n\\cite[Proposition~1.14]{CaFe11} to go through: we always have\n\\begin{align*}\n   \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps(t)\\|_{L^2}& \\\\\n\\lesssim\n  \\sum_{{j\\not =k, \\ \\ell_j,\\ell_k\\ge 1}\\atop {\\ell_j+\\ell_k+\\ell_m =3}}&\\left\\|\n  u_j^{\\ell_j}\\(t,y-\\frac{q_j(t)-q_k(t)}{\\sqrt\\eps } \\)\n  u_k^{\\ell_k}(t,y)u_m^{\\ell_m}\\(t,y-\\frac{q_m(t)-q_k(t)}{\\sqrt\\eps } \\)\\right\\|_{L^2}\\\\\n\\lesssim\n  \\sum_{{j\\not =k,\\  \\ell_j,\\ell_k\\ge 1}\\atop {\\ell_j+\\ell_k+\\ell_m\n  =3}}&\\|u_m(t)\\|_{L^\\infty}^{\\ell_m}\\left\\| \n  u_j^{\\ell_j}\\(t,y-\\frac{q_j(t)-q_k(t)}{\\sqrt\\eps } \\)\n  u_k^{\\ell_k}(t,y)\\right\\|_{L^2},\n\\end{align*}\nso the last factor is exactly the one considered in \\cite{CaFe11} and\n  above. \n\\subsection*{Acknowledgements}\nThe author is grateful to Jean-Fran\\c cois Bony, Clotilde Fermanian,\nIsabelle Gallagher and Fabricio Maci\\`a for fruitful discussions about\nthis work. \n\n\\bibliographystyle{siam}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}