\documentclass[12pt,reqno]{amsart} \usepackage{amsmath,amsthm,amscd,amsfonts,amssymb,graphicx,color} \usepackage[bookmarksnumbered,colorlinks,plainpages]{hyperref} \hypersetup{colorlinks=true,linkcolor=red,anchorcolor=green,citecolor=cyan,urlcolor=red,filecolor=magenta,pdftoolbar=true} \usepackage{mathrsfs} % ---------------------------------------- \textheight 22.5truecm \textwidth 14.5truecm \setlength{\oddsidemargin}{0.35in} \setlength{\evensidemargin}{0.35in} \setlength{\topmargin}{-.5cm} % ---------------------------------------- \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{notation}[theorem]{Notation} \numberwithin{equation}{section} % ------------------------------------------- \usepackage{microtype} \begin{document} \pdfcompresslevel=0 \pdfobjcompresslevel=0 \pdfinterwordspaceon \setcounter{page}{1} \title[A phase space localization operator in negative binomial states]{ A phase space localization operator in negative binomial states} \author[Zouha\"{\i}r Mouayn, Soumia Touhami ]{ Zouha\"{\i}r Mouayn $^{1,2,3}$, Soumia Touhami $^4$} % ----- \address{\footnotesize $^1$ Department of Mathematics, Faculty of Sciences and Technics (M'Ghila), {\scriptsize Sultan Moulay Slimane University, B\'{e}ni Mellal, Morocco.}} \address{\footnotesize $^2$ Institut des Hautes Études Scientifiques, Paris Saclay University Le Bois-Marie, {\scriptsize 35 route de Chartres CS 40001 91893 Bures-sur-Yvette, France.}} \address{\footnotesize $^3$ Institut Henri Poincaré - UAR 839 Sorbonne University, / CNRS, {\scriptsize 11 rue Pierre et Marie Curie 75231 Paris Cedex 05 France.}} \address{\footnotesize $^4$ Department of Mathematics, KTH Royal Institute of Technology, {\scriptsize Stockholm, Sweden.}} \email{\textcolor[rgb]{0.00,0.00,0.84}{mouayn@gmail.com $^{1}$; touhami16soumia@gmail.com $^{2}$}} % ----------------------------------- \keywords{} % ----------------------------------- \begin{abstract} We are dealing with some spectral properties of a phase space localization operator $P_{R}$ corresponding to the indicator function of a disk of radius $R<1.$ The localization procedure is achieved with respect to a set of negative binomial states (NBS) labeled by points of the complex unit disk\ $% \mathbb{D}$ and depending on a parameter $B>% %TCIMACRO{\U{bd}}% %BeginExpansion {\frac12}% %EndExpansion $ $.$ We derive a formula expressing $P_{R}$ as function of the pseudo-harmonic oscillator whose potential function depends on $B$. The phase space content outside the localization domain is estimated in terms of the photon-counting probability distribution associated with the NBS. By using the coherent states transform attached to NBS, we transfer the action of the operator $P_{R}$ to a Bergman space\ $\mathcal{A}^{B}\left( \mathbb{D}% \right) $ of analytic functions on\ $\mathbb{D}$ satisfying a growth condition depending on $B$ and we explicitly give its integral kernel whose limit\ as $R\rightarrow 1$ coincides with the reproducing kernel of \ $% \mathcal{A}^{B}\left( \mathbb{D}\right) $. This leads to a natural generalization of this Hilbert space with respect to the parameter $R$. . \end{abstract} \maketitle % ------------------------------------------- \section{\ \textbf{Introduction}} The problem of localization in time and frequency has always been of serious concern in modern physics because one of the major issues in applications is to analyze signals on different time-frequency domains and therefore to concentrate and localize signals on these domains. To be able to represent the frequency behavior of a signal locally in time, one has to consider the so called time-frequency localization operators. A variety of methods have been invented to construct such class of operators \cite{W}. Coherent states (CS) are the natural tool in constructing phase space localization operators \ and have been extensively encountered in theoretical physics, in quantum mechanics and in many different areas of mathematical physics. Precisely, CS provide a close connection between classical and quantum formalisms so as to play a central role in the semi classical analysis. In general, they may be defined as an overcomplete family of normalized ket vectors $|\zeta \rangle $ which are labeled by points $\zeta $ of a phase-space domain $X$, belonging to a Hilbert space $% \mathcal{H}$ that corresponds to a specific quantum model and provide $% \mathcal{H}$ with a resolution of its identity operator as \begin{equation} \label{r1} \mathbf{1}_{\mathcal{H}}=\int_{X}|\zeta \rangle \langle \zeta |d\mu (\zeta ). \end{equation} with respect to a suitable integration measure $d\mu (\zeta )$ on $X$. These states are constructed in different ways. For an overview of all aspects of the theory of coherent states and their genesis, we refer to the \cite{V,TP}. \smallskip Equation $\eqref{r1} $ allows to implement a CS frame quantization \cite{G} of the set of parameters $\zeta \in X$ by associating to a complex-valued function $\zeta \mapsto F(\zeta )$, satisfying appropriate conditions, the following operator on $\mathcal{H}$ : \begin{equation} \label{r2} F(\zeta )\mapsto P_{F}:=\int\limits_{X}|\zeta \rangle \langle \zeta |F(\zeta )d\mu (\zeta ). \end{equation} If $F(\zeta )$ is semi-bounded real-valued function, the Friedrich extension \cite{RS} allows us to define $P_{F}$ as a self-adjoint operator. In particular, when $F=\chi _{\Omega }$ is the indicator function for some domain $\Omega $ in the phase space $X$, the resulting operator $P_{\chi _{\Omega }}$ is called a localization operator. \smallskip \smallskip By using the CS of the harmonic oscillator, Daubechies \cite{Daub} has discussed the localization operator $P_{\chi _{\Omega }}$ with $\Omega \subset \mathbb{C}$ being a disk of radius $\rho >0$ by giving its eigenfunctions in terms of Hermite polynomials, and by expressing its discrete eigenvalues $\left\{ \lambda _{k}^{\rho }\right\} $ in term of incomplete Gamma functions. She also has established the asymptotic behavior of these eigenvalues for varying $k=0,1,2,...,$ and $\rho >0$, and has given an estimate for the phase-content outside the localization domain $\Omega $. \smallskip In this paper, we deal with similar questions for the pseudo-harmonic oscillator \begin{equation} \label{r3} H_{B}=\frac{1}{2}\left[ -\frac{d^{2}}{dx^{2}}+x^{2}+\frac{(2B-1)^{2}-\frac{1% }{4}}{x^{2}}\right] +\left( 1-B\right) ,\text{ \ \ \ \ \ }2B>1 \end{equation} acting on the Hilbert space $L^{2}\left( \mathbb{R}_{+}\right) $, whose importance consists in the fact that it is a solvable model and being, in a certain sense, an intermediate potential between the three dimensional harmonic oscillator potential and other anharmonic potentials such as Poschl-Teller or Morse potential \cite{PD1, PD2}. The $L^{2}$ eigenfunctions (number states) of $H_{B}$, which here are denoted by the ket vectors $\left\vert \ell _{j}^{B}\right\rangle ,$ \ may be superposed to perform a set of coherent states within the so-called \textit{Hilbertian probabilistic scheme} (see \cite{G}for the general theory) by choosing a set of analytic coefficients $C_{j}^{B}\left( z\right) $ on the complex unit disk $\mathbb{D=}\left\{ z\in \mathbb{C},\left\vert z\right\vert <1\right\} $ such that the associated photon-counting statistics follows a negative probability distribution. Such a CS are known as the negative binomials states (NBS) \cite{Mou1}.\ One interest on them is that they intermediate between pure coherent states and pure thermal states \cite{GJT}and reduce to Susskind-Glogower phases states for a particular limit of the parameter \cite{FS} Beside, such coefficients $C_{j}^{B}\left( z\right) $, turn out to be basis elements of the weighted Bergman space, here denoted $\mathcal{A}^{B}\left( \mathbb{D}% \right) $, of analytic functions $g$ on $\mathbb{D}$, satisfying the growth condition $\int_{\mathbb{D}}|g(z)|^{2}(1-\bar{z}z)^{2B-2}d\eta (z)<+\infty $% , where $d\eta $ denotes the Lebesgue measure on $\mathbb{D}$. Our aim is, firstly, to show that these NBS which are labeled by points of the disk $\mathbb{D}$ can be retreived from the affine CS via the Cayley transform. We also link them to the Landau problem in the Poincar\'{e} upper half-plane. This connection may be exploited to generalize the obtained results to higher hyperbolic Landau levels. Secondly, we proceed by a quantization method based on these NBS in order to construct a phase space localization operator $P_{R}$ corresponding to the disk $D_{R}=\left\{ z\in \mathbb{C},\left\vert z\right\vert 0\text{.} \end{equation} This representation is square integrable since it is easy to find a vector $% \phi _{0}\in \mathcal{H}$ such that the function $\left( x,y\right) \mapsto \left\langle \pi _{+}\left( x,y\right) \left[ \phi _{0}\right] ,\phi _{0}\right\rangle _{\mathcal{H}}$ belongs to $L^{2}\left( \mathbf{G},d\nu \right) $. This condition can also be expressed by saying that the self-adjoint operator $\delta :\mathcal{H\rightarrow H}$ defined as $\delta % \left[ \varphi \right] (\xi )=\xi ^{-\frac{1}{2}}\varphi \left( \xi \right) $ gives \begin{equation} \label{r5} \int\limits_{\mathbf{G}}\left\langle \varphi _{1},\pi _{+}\left( x,y\right) % \left[ \psi _{1}\right] \right\rangle \left\langle \pi _{+}\left( x,y\right) % \left[ \varphi _{2}\right] ,\psi _{2}\right\rangle d\nu \left( x,y\right) =\left\langle \varphi _{1},\varphi _{2}\right\rangle \left\langle \delta ^{% \frac{1}{2}}\left[ \varphi _{1}\right] ,\delta ^{\frac{1}{2}}\left[ \varphi _{2}\right] \right\rangle \end{equation} for all $\psi _{1},\psi _{2},\varphi _{1},\varphi _{2}\in \mathcal{H}$. The operator $\delta $ is unbounded because $\mathbf{G}$ is not unimodular \cite{DMo}. \smallskip Keeping the condition $2B>1,$ we consider a set of CS labeled by elements $% (x,y)\in \mathbf{G}$, which are obtained by acting, via the representation operator $\pi _{+}\left( x,y\right) $, on the admissible vector \begin{equation} \label{r6} \phi _{B}\left( \xi \right) :=\frac{1}{\sqrt{2B}}\xi ^{B}e^{-\frac{1}{2}\xi },\text{ \ \ }\xi >0. \end{equation} Precisely, \begin{equation} \label{r7} \left\vert \tau _{(x,y),B}\right\rangle :=\pi _{+}\left( x,y\right) \left[ \phi _{B}\right] \end{equation} and satisfy the resolution of the identity operator \begin{equation} \label{r8} \mathbf{1}_{\mathcal{H}}=c_{B}\int\limits_{\mathbf{G}}d\mu \left( x,y\right) \left\vert \tau _{(x,y),B}\right\rangle \left\langle \tau _{(x,y),B}\right\vert \end{equation}% where $c_{B}:=2B-1$ and the Dirac's bra-ket notation $|\Phi \rangle \langle \Phi |$ means the rank-one operator $\phi \longmapsto \langle \Phi ,\phi \rangle _{\mathcal{H}}.\Phi $ with $\Phi ,\phi \in \mathcal{H}$. In the $\xi $-coordinate, wavefunctions of CS defined by Eq. $\eqref{r7} $ read \begin{equation} \label{r9} \left\langle \xi \right\vert \tau _{(x,y),B}\rangle =\frac{1}{\sqrt{2B}}% \left( \xi y\right) ^{B}e^{-\frac{1}{2}\xi \left( y-ix\right) }\text{, }% \qquad \xi >0, \end{equation} and are known as the affine CS \cite{AK}. \subsection{Connection with the $B$-weight Maass Laplacian} To describe the connection of CS $\eqref{r9} $ with the lowest hyperbolic Landau level, we may first identify the affine group $\mathbf{G}$ with the Poincar\'{e} upper half-plane $\mathbb{H}^{2}=\left\{ x+iy,x\in \mathbb{R},y>0\right\} $. Then, to these CS we may attach, as usual, the CS transform $\mathcal{B}_{0}:\mathcal{H}\rightarrow L^{2}\left( \mathbb{H}% ^{2},d\nu \right) $ defined by \cite{Mou2}: \begin{equation} \label{r10} \mathcal{B}_{0}[\phi ]\left( x,y\right) =\sqrt{c_{B}}\int\limits_{0}^{+% \infty }\overline{\left\langle \xi \right\vert \tau _{(x,y),B}\rangle }\phi (\xi )\xi ^{-1}d\xi \end{equation} whose range is the eigenspace of the $B$-weight Maass Laplacian \begin{equation} \Delta _{B}=y^{2}\left( \partial _{x}^{2}+\partial _{y}^{2}\right) -2iBy\partial _{x}, \end{equation} associated with the eigenvalue \begin{equation} \label{r11} \epsilon _{m}^{B}=(B-m)\left( 1-B+m\right) ,\quad m=0,1,...,\left\lfloor B-% %TCIMACRO{\U{bd}}% %BeginExpansion {\frac12}% %EndExpansion \right\rfloor , \end{equation} where $\left\lfloor a\right\rfloor $ denotes the greatest integer not exceeding $a.$ We precisely have \begin{equation} \label{r12} \mathcal{B}_{0}[\mathcal{H}]\equiv \left\{ f\in L^{2}\left( \mathbb{H}% ^{2},d\nu \right) ,\;\Delta _{B}f=\epsilon _{0}^{B}f\right\} . \end{equation} The operator $\Delta _{B}$ also stands (in suitable units and up to an additive constant) for the Schr\"{o}dinger operator describing the dynamics of a charged particle moving on $\mathbb{H}^{2}$ under the action of a magnetic field of strength proportional to $B.$ This is an elliptic densely defined operator on the Hilbert space $L^{2}(\mathbb{H}^{2},d\nu )$, with a unique self-adjoint realization also denoted by $\Delta _{B}$. Its spectrum consists of two parts:\textit{\ }a continuous part $\left[ %TCIMACRO{\U{bc}}% %BeginExpansion {\frac14}% %EndExpansion \theta ,+\infty \right[ $, corresponding to scattering states and the finite number of eigenvalues $\epsilon _{m}^{B}$ each one with infinite degeneracy,\ called hyperbolic Landau levels . Finally, the reproducing kernel of the Hilbert space $\mathcal{B}_{0}[\mathcal{H}]$ can be obtained from the overlapping function $\langle \tau _{w,B},\tau _{\zeta ,B}\rangle _{% \mathcal{H}}$ between two CS as \begin{equation} \label{r13} K_{0}^{B}\left( w,\zeta \right) =\left( \frac{\left\vert w-\bar{\zeta}\right\vert ^{2}}{4\rm{Im}w\rm{Im}\zeta }\right) ^{-B}\left( \frac{\zeta-\bar{w}}{w-\bar{\zeta}}\right) ^{B}{},\text{ \ }w,\zeta \in \mathbb{H}^{2}. \end{equation} \subsection{Negative binomial states} \smallskip We can write a version of these CS as states labeled by points $z$ of the unit disk $\mathbb{D}$ by using the inverse Cayley transform $% \mathcal{C}^{-1}:\mathbb{D}\rightarrow \mathbf{G}$ given by \begin{equation} \label{r14} \mathcal{C}^{-1}\left( z\right) =\left( -2\frac{\rm{Im}z}{\left\vert 1-z\right\vert ^{2}},\frac{1-\left\vert z\right\vert ^{2}}{\left\vert 1-z\right\vert ^{2}}\right) ,\text{ \ }z\in \mathbb{D}. \end{equation} Indeed, we may still define this version as states in $\mathcal{H}$ as% \begin{equation} \label{r15} \kappa _{z,B}:=\left( \frac{1-\bar{z}}{1-z}\right) ^{B}\pi _{+}\left( \mathcal{C}^{-1}\left( z\right) \right) \left[ \phi _{B}\right] . \end{equation} Direct calculations lead to their wave functions in the $\xi $-coordinate as \begin{equation} \label{r16} \left\langle \xi \right\vert \kappa _{z,B}\rangle =\frac{1}{\sqrt{\Gamma \left( 2B\right) }}\left( \frac{\left( 1-z\bar{z}\right) }{\left( 1-z\right) ^{2}}\xi \right) ^{B}\exp \left( -\frac{1}{2}\left( \frac{1+z}{1-z}\right) \xi \right) ,\qquad \xi >0. \end{equation} The latter ones may slightly be modified in order to perform them as vectors of $L^{2}(\mathbb{R}_{+},d\xi )$ labeled by points of the disk $\mathbb{D}$ as \begin{equation} \label{r17} \left\langle \xi \right\vert \widetilde{\kappa }_{z,B}\rangle :=\sqrt{\frac{2% }{\xi }}\,\langle \xi ^{2}|\kappa _{z,B}\rangle \end{equation} These states obey the normalization condition $\langle \widetilde{\kappa }% _{z,B},\widetilde{\kappa }_{z,B}\rangle _{L^{2}(\mathbb{R}_{+})}=1$ and satisfy the resolution of the identity operator as \begin{equation} \label{r18} 1_{L^{2}(\mathbb{R}_{+})}=\int\limits_{\mathbb{D}}\left\vert \widetilde{% \kappa }_{z,B}\right\rangle \left\langle \widetilde{\kappa }% _{z,B}\right\vert d\eta _{B}(z), \end{equation} with respect to the measure \begin{equation} \label{r19} d\eta _{B}(z):=\frac{(2B-1)}{\pi \left( 1-z\overline{z}\right) ^{2}}d\eta (z). \end{equation} Now, as for the canonical CS of the harmonic oscillator, we seek for a number states expansion of the CS $\left\vert \widetilde{\kappa }% _{z,B}\right\rangle $ in terms of the analytic coefficients \begin{equation} \label{r20} C_{j}^{B}(z):=(2B-1)^{1/2}\sqrt{\frac{\Gamma (2B+j)}{j!\Gamma (2B)}}z^{j},% \text{ \ \ }j=0,1,2,..., \end{equation} which constitute an orthonormal basis of the weighted Bergman space $% \mathcal{A}^{B}(\mathbb{D})$. For that, we start from the expression of the CS, $\left\vert \widetilde{\kappa }_{z,B}\right\rangle $ as given by $\eqref{r17}-\eqref{r16}$ and we make use of the generating function for the Laguerre polynomials (\cite{MaOb}, p.239): \begin{equation} \label{r21} \sum_{j=0}^{+\infty }t^{j}L_{j}^{(\alpha )}(x)=\frac{1}{(1-t)^{\alpha +1}}% \exp \left( \frac{-t}{1-t}x\right) ,\ \ \alpha >-1. \end{equation} We obtain, after some calculations, the series expansion \begin{equation} \label{r22} \langle \xi \left\vert \widetilde{\kappa }_{z,B}\right\rangle =\left( \frac{% 2B-1}{(1-z\bar{z})^{2B}}\right) ^{-1/2}\sum_{j=0}^{+\infty }C_{j}^{B}(z)\ell _{j}^{B}(\xi ) \end{equation} where $\ell _{j}^{B}(\xi )$ are the Laguerre functions \begin{equation} \label{r23} \ell _{j}^{B}(\xi ):=\left( \frac{2j!}{\Gamma (2B+j)}\right) ^{1/2}\xi ^{2B-% \frac{1}{2}}e^{-\frac{1}{2}\xi ^{2}}L_{j}^{(2B-1)}(\xi ^{2}),\text{ \ \ }% j=0,1,2,..., \end{equation} which are known to constitute a complete orthonormal system in $L^{2}(% \mathbb{R}_{+},d\xi )$. \smallskip \begin{remark} The CS $\eqref{r16} $ also coincide with those constructed by Molanar \textit{et al} \cite{BMMB} for the Morse potential \cite{MP} by an algebraic way based on supersymetry and shape invariance properties where the shape parameter may be taken as our $% B>0$. For this potential, they first were were introduced by Nieto \textit{% et al} \cite{NMSL} as generalized minimal uncertainty states. \end{remark} \section{{\ \textbf{Quantization via $\ $NBS $\left\vert \widetilde{\protect\kappa }% _{z,B}\right\rangle $}}} The resolution of the identity $\eqref{r18}$ allows to implement a CS or frame quantization \cite{TP} of the set of parameters $% \mathbb{D}$ by associating to a function $\mathbb{D}\ni z\mapsto F(z,% \overline{z})\in \mathbb{C}$ that satisfies appropriate conditions, the following operator in $L^{2}(\mathbb{R}_{+},d\xi ):$ \begin{equation} \label{r24} F\mapsto \wp _{F}^{B}:=\int\limits_{\mathbb{D}}\left\vert \widetilde{\kappa }% _{z,B}\right\rangle \left\langle \widetilde{\kappa }_{z,B}\right\vert F(z,% \overline{z})\frac{(2B-1)}{\pi (1-z\bar{z})^{2}}d\eta (z). \end{equation} The Friedrich extension \cite{RS} allows to define $\wp _{F}^{B}$ as a self-adjoint operator if $F$ is a semi-bounded real-valued function.% \newline \bigskip \smallskip In order to proceed with the quantization through the CS $\left\vert \widetilde{\kappa }_{z,B}\right\rangle $ along the linear map $\eqref{r24} $, we may substitute the expression $\eqref{r22}$ into the integral form in $\eqref{r24} .$ This gives the expression% \begin{equation} \label{r25} \wp _{F}^{B}=\sum_{j,k=0}^{+\infty }\sqrt{\frac{\Gamma (2B+j)}{\pi j!\Gamma (2B)}}\sqrt{\frac{\Gamma (2B+k)}{\pi k!\Gamma (2B)}}\left[ (2B-1)\int\limits_{\mathbb{D}}z^{k}\bar{z}^{j}(1-z\bar{z})^{2B-2}F(z,\bar{z}% )d\eta (z)\right] \left\vert \ell _{j}^{B}\right\rangle \left\langle \ell _{k}^{B}\right\vert \end{equation} which represents its disrete spectral resolution\textit{\ }% \begin{equation} \label{r26} \wp _{F}^{B}=\sum_{j,k=0}^{+\infty }\left[ \gamma _{F}^{B}\right] _{j,k}\left\vert \ell _{k}^{B}\rangle \langle \ell _{j}^{B}\right\vert \end{equation} where the matrix elements are (at least formally) given by\textit{\ }% \begin{equation} \label{r27} \left[ \gamma _{F}^{B}\right] _{j,k}=\frac{1}{\pi \Gamma (2B-1)}\left( \frac{% \Gamma (2B+j)\Gamma (2B+k)}{j!k!}\right) ^{1/2}\int\limits_{\mathbb{D}}\bar{z% }^{j}z^{k}(1-z\bar{z})^{2B-2}F(z,\bar{z})d\eta (z) \end{equation} and\ $\left\{ \ell _{j}^{B}\right\} $ is the orthonormal basis of $L^{2}(% \mathbb{R}_{+},d\xi ),$\ which is given in $\eqref{r23}.$ \subsection{Radial classical observables} For a radial weight function $F$, the above discrete spectral resolution of $% \wp _{F}^{B}$ leads to more precise expressions for its eigenvalues. Indeed, by setting $F(z,\bar{z})=F(r^{2})$, $r=\left\vert z\right\vert $ and using polar coordinates in the expression $\eqref{r27} $ of matrix elements, we get that \begin{equation} \label{r28} \left[ \gamma _{F}^{B}\right] _{j,k}=\frac{1}{\pi \Gamma (2B-1)}\left( \frac{% \Gamma (2B+j)\Gamma (2B+k)}{j!k!}\right) ^{1/2}\int\limits_{0}^{2\pi }\int\limits_{0}^{1}r^{k}r^{j}e^{ik\theta }e^{-ij\theta }(1-r^{2})^{2B-2}F(r^{2})rdrd\theta . \end{equation} By the fact that \begin{equation} \label{r29} \int\limits_{0}^{2\pi }e^{i(k-j)\theta }d\theta =2\pi \delta _{k,j},\text{ \ }j,k=0,1,2,..., \end{equation} on can easily see that only the case $j=k$ produces a non zero matrix element as \begin{equation} \label{r30} \left[ \gamma _{F}^{B}\right] _{j,j}=\frac{\Gamma (2B+j)}{\Gamma (2B-1)\Gamma (j+1)}\int_{0}^{1}\rho ^{j}(1-\rho )^{2B-2}F(\rho )d\rho . \end{equation} By writing the prefactor as $\left( \mathcal{B}(j+1,2B-1)\right) ^{-1}$, we obtain the expression of the $\lambda _{j}^{B,F}$ as \begin{equation} \label{r31} \lambda _{j}^{B,F}=\frac{1}{\mathcal{B}(j+1,2B-1)}\int\limits_{0}^{1}\rho ^{j}(1-\rho )^{2B-2}F(\rho )d\rho , \end{equation} where $\mathcal{B}(a,b)$ denotes the Beta function with $a,b>0$. The operator $\wp _{F}^{B}$ has the following discrete spectral resolution with respect to the orthonormal basis $\{\ell _{j}^{B}\}$ as \begin{equation} \label{r32} \wp _{F}^{B}=\sum_{j=0}^{+\infty }\lambda _{j}^{B,F}\left\vert \ell _{j}^{B}\right\rangle \left\langle \ell _{j}^{B}\right\vert , \end{equation} and it's not difficult to check that \begin{equation} \label{r33} \wp _{F}^{B}\left[ \ell _{j}^{B}\right] =\lambda _{j}^{B,F}\ell _{j}^{B}. \end{equation} Note that $\eqref{r33} $ means that the operators $\wp _{F}^{B}$ and $% H_{B}$ have $\{\ell _{j}^{B}\}$ as a commun set of eigenfunctions. \smallskip In particular, we here consider the disk $D_{R}:=\{z\in \mathbb{C},\ |z|1$ and $m=0$ this eigenspace reduces to the weighted Bergman space of analytic functions $g$ on $\mathbb{D}$, satisfying the growth condition $\int_{\mathbb{D}% }|g(z)|^{2}(1-\bar{z}z)^{2B-2}d\eta (z)<+\infty $. In other words, $\mathcal{% E}_{B,0}\left( \mathbb{D}\right) \equiv \mathcal{A}^{B}\left( \mathbb{D}% \right) .$ This remark makes possible to extend our analysis to the above higher hyperbolic Landau levels $\sigma _{B,m}.$ In this respect the connection with the results in \cite{CDM} may be useful. \section{{\ \textbf{Phase space content of $\wp _{R}^{B}$ outside $D_{R}$}}} Note that the phase space cutoff by the operator $\wp _{R}^{B}$ is not sharp in the sense that it will have some phase space content outside the localization domain $D_{R}$. This is illustrated by the fact that at least for some coherence point $z_{0}\in \mathbb{D}\setminus D_{R}$ we have \begin{equation} \label{r47} \langle \widetilde{\kappa }_{z_{0},B}\left\vert \wp _{R}^{B}[f]\right\rangle _{L^{2}(\mathbb{R}_{+})}\neq 0,\quad f\in L^{2}(\mathbb{R}_{+}). \end{equation} Precisely, we have the following estimate involving the photon counting statistics which obey the negative binomial probability distribution $% \mathcal{X}$ $\sim \mathcal{NB}(2B,z_{0}\bar{z}_{0})$\textit{\ }with\textit{% \ }parameters $2B$\ and $z_{0}\bar{z}_{0}$ \begin{equation} \label{r48} \left\vert \langle \widetilde{\kappa }_{z_{0},B}\left\vert \wp _{R}^{B}[f]\right\rangle _{L^{2}\left( \mathbb{R}_{+}\right) }\right\vert \leq \sqrt{\mathbb{E}\left( \left( \mathcal{I}_{R^{2}}\left( \mathcal{X+}% 1,2B-1\right) \right) ^{2}\right) }\Vert f\Vert _{L^{2}\left( \mathbb{R}% _{+}\right) } . \end{equation} To prove $\eqref{r48}$ we start by replacing in the scalar product $\eqref{r47}$ the operator $\wp _{R}^{B}$ by its discrete spectral resolution $\eqref{r36}$ as \begin{equation} \label{r49} \left\langle \widetilde{\kappa }_{z_{0},B},\left( \sum_{j=0}^{+\infty }\lambda _{j}^{B,R}\left\vert \ell _{j}^{B}\right\rangle \left\langle \ell _{j}^{B}\right\vert \right) [f]\right\rangle _{L^{2}(\mathbb{R}% _{+})}=\sum_{j=0}^{+\infty }\lambda _{j}^{B,R}\left\langle \widetilde{\kappa }_{z_{0},B}|\ell _{j}^{B}\right\rangle _{L^{2}\left( \mathbb{R}_{+}\right) }\left\langle \ell _{j}^{B}|f\right\rangle _{L^{2}\left( \mathbb{R}% _{+}\right) }. \end{equation} Recalling the number states expansion $\eqref{r22} $ of $\left\vert \widetilde{\kappa }_{z,B}\right\rangle $, we may write \begin{equation} \label{r50} \left\langle \widetilde{\kappa }_{z_{0},B}|\ell _{j}^{B}\right\rangle _{L^{2}\left( \mathbb{R}_{+}\right) }=(1-z_{0}\bar{z}_{0})^{B}\sqrt{\frac{% \Gamma (2B+j)}{j!\Gamma (2B)}}z_{0}^{j}. \end{equation} Therefore $\eqref{r49} $ reads \begin{equation} \label{r51} \langle \widetilde{\kappa }_{z_{0},B}\left\vert \wp _{R}^{B}[f]\right\rangle _{L^{2}\left( \mathbb{R}_{+}\right) }=(1-z_{0}\bar{z_{0}})^{B}\sum_{j=0}^{+% \infty }\lambda _{j}^{B,R}\left( \frac{\Gamma (2B+j)}{j!\Gamma (2B)}\right) ^{1/2}\left\langle \ell _{j}^{B}|f\right\rangle _{L^{2}\left( \mathbb{R}% _{+}\right) }z_{0}^{j}. \end{equation} \begin{equation} \label{r52} =(1-z_{0}\bar{z_{0}})^{B}\left\langle \sum_{j=0}^{+\infty }\lambda _{j}^{B,R}\left( \frac{\Gamma (2B+j)}{j!\Gamma (2B)}\right) ^{1/2}z_{0}^{j}\ell _{j}^{B}|f\right\rangle _{L^{2}(\mathbb{R}_{+})}. \end{equation} Setting \begin{equation} \label{r53} \vartheta :=\sum_{j=0}^{+\infty }\lambda _{j}^{B,R}\left( \frac{\Gamma (2B+j)% }{j!\Gamma (2B)}\right) ^{1/2}z_{0}^{j}\ell _{j}^{B}, \end{equation} and using the Cauchy-Bunyakovsky-Schwarz inequality \begin{equation} \label{r54} \left\vert \langle \widetilde{\kappa }_{z_{0},B}|\wp _{R}^{B}[f]\rangle _{L^{2}\left( \mathbb{R}_{+}\right) }\right\vert \leq |1-z_{0}\bar{z_{0}}% |^{B}\Vert \vartheta \Vert _{L^{2}(\mathbb{R}_{+})}\left\Vert f\right\Vert _{L^{2}\left( \mathbb{R}_{+}\right) }. \end{equation} Now, the $L^{2}$ norm of $\vartheta $ may be written \begin{equation} \label{r54} |1-z_{0}\bar{z_{0}}|^{B}\Vert \vartheta \Vert _{L^{2}(\mathbb{R}% _{+})}=\left( \sum_{j=0}^{+\infty }\left( \lambda _{j}^{B,R}\right) ^{2}% \left[ \frac{\Gamma (2B+j)}{j!\Gamma (2B)}(z_{0}\bar{z_{0}})^{j}(1-z_{0}\bar{% z_{0}})^{2B}\right] \right) ^{1/2}. \end{equation} By recognizing in $\eqref{r54} $ the probability distribution of $% \mathcal{X}\sim \mathcal{N}\mathcal{B}(2B,z_{0}\overline{z_{0}})$ : \begin{equation} \label{r55} \Pr \left( \mathcal{X}=j\right) =\frac{\Gamma (2B+j)}{j!\Gamma (2B)}\left( z_{0}\overline{z_{0}}\right) ^{j}(1-z_{0}\overline{z_{0}})^{2B},\ \ j=0,1,2,\cdots , \end{equation} and using the expression of the eigenvalues $\lambda _{j}^{B,R}$, Eq.$\eqref{r54} $ takes the form \begin{equation} \label{r56} |1-z_{0}\bar{z_{0}}|^{B}\Vert \vartheta \Vert _{L^{2}(\mathbb{R}% _{+})}=\left( \sum_{j=0}^{+\infty }\left( \mathcal{I}_{R^{2}}(j+1,2B-1)% \right) ^{2}\Pr \left( \mathcal{X}=j\right) \right) ^{1/2}. \end{equation} Finally, the right hand side of $\eqref{r56} $ can be viewed as the square root of an expectation value as $\mathbb{E}\left( \left( \mathcal{I}% _{R^{2}}(\mathcal{X}+1,2B-1)\right) ^{2}\right) $. \section{\textbf{Integral kernel of $\widetilde{\wp }_{R}^{B}$ \ on the Bergman space $\mathcal{A}^{B}(\mathbb{D})$}} The CS transform associated with $\left\vert \widetilde{\kappa }% _{z,B}\right\rangle $ is the isometric isomorphism $W_{B}:L^{2}(\mathbb{R}% _{+})\rightarrow \mathcal{A}^{B}(\mathbb{D})$ defined by $\varphi \mapsto W_{B}[\varphi ](z):=(1-z\bar{z})^{-B}\langle \varphi \left\vert \widetilde{% \kappa }_{z,B}\right\rangle _{L^{2}\left( \mathbb{R}_{+}\right) }$, which may also be called the second Bargmann transform \cite{IWGM}. Explicitly, \begin{equation} \label{r5.1} W_{B}[f](z)=\sqrt{\frac{2}{\Gamma (2B)}}(1-z)^{-2B}\int_{0}^{+\infty }\xi ^{2B-\frac{1}{2}}\exp \left( -\frac{1}{2}\left( \frac{1+z}{1-z}\right) \xi ^{2}\right) d\xi ,\ z\in \mathbb{D}. \end{equation} Now, to transfer the operator $\wp _{R}^{B}$ to be acting on functions $f$ $% \in $ $\mathcal{A}^{B}(\mathbb{D}),$ we use the relation $W_{B}\circ \wp _{R}^{B}\circ W_{B}^{-1}\equiv \widetilde{\wp }_{R}^{B}$. \ We, precisely, obtain that \begin{equation} \label{r5.2} \widetilde{\wp }_{R}^{B}[f](w)=\int\limits_{\mathbb{D}}P_{R}^{B}(z,w)f(z)(1-z% \bar{z})^{2B-2}d\eta (z) \end{equation} where the integral kernel is given by \begin{equation} \label{r5.3} P_{R}^{B}(z,w)=\frac{(2B-1)^{2}R}{(1-R\bar{z}w)^{2B}}F_{1}\left( 2-2B,1-2B,2B,2,R,\frac{R-R\bar{z}w}{1-R\bar{z}w}\right) \end{equation} In particular, at the limit $R\rightarrow 1$, \begin{equation} \label{r5.4} \lim_{R\rightarrow 1}P_{R}^{B}(z,w)=\frac{(2B-1)}{\left( 1-\bar{z}w\right) ^{2B}}, \end{equation} which is the reproducing kernel of the Bergman space $\mathcal{A}^{B}(% \mathbb{D}).$ Here, $F_{1}$ denotes the Appel hypergeometric double series% \begin{equation*} F_{1}\left( \alpha ,\beta ,\gamma ;\omega ;u,v\right) =\sum\limits_{j,k=0}^{\infty }\frac{\left( \alpha \right) _{j+k}\left( \beta \right) _{j}\left( \gamma \right) _{k}}{j!k!\left( \omega \right) _{j+k}}% u^{j}v^{k},\left\vert u\right\vert <1,\left\vert v\right\vert <1. \end{equation*} \bigskip \smallskip To prove $\eqref{r5.2} ,$ let us take $f\in \mathcal{A}^{B}(\mathbb{D})$ and apply the inverse of $W_{B}$ as \begin{equation} \label{r5.5} W_{B}^{-1}[f](\xi )=\int\limits_{\mathbb{D}}f(z)\left\langle \xi \right\vert \widetilde{\kappa }_{z,B}\rangle (1-z\bar{z})^{-B}d\eta _{B}(z),\text{ }\xi >0. \end{equation} Next, we proceed by the action of $\wp _{R}^{B}$ on $W_{B}^{-1}[f],$ which successively gives \begin{equation} \label{r5.6} \wp _{R}^{B}\left[ W_{B}^{-1}[f]\right] (y)=\sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\left\langle \ell _{k}^{B}|\int\limits_{\mathbb{D}}f(z)|\widetilde{% \kappa }_{z,B}\rangle (1-z\bar{z})^{-B}d\eta _{B}(z)\right\rangle \left\langle y|\ell _{k}^{B}\right\rangle \end{equation} \begin{equation} \label{r5.7} =\sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\left( \int_{\mathbb{R}_{+}}% \overline{\ell _{k}^{B}(\xi )}\left( \int_{\mathbb{D}}f(z)\left\langle x\right\vert \widetilde{\kappa }_{z,B}\rangle (1-z\bar{z})^{-B}d\eta _{B}(z)\right) d\xi \right) \ell _{k}^{B}(y) \end{equation} \begin{equation} \label{r5.8} =\sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\int_{\mathbb{D}}f(z)\left( (1-z\bar{% z})^{-B}\overline{\int_{\mathbb{R}_{+}}\ell _{k}^{B}(\xi )\overline{% \left\langle \xi \right\vert \widetilde{\kappa }_{z,B}\rangle }}dx\right) d\eta _{B}\left( z\right) \ell _{k}^{B}(y). \end{equation} Note that the last equation may be rewritten as \begin{equation} \label{r5.9} \wp _{R}^{B}\left[ W_{B}^{-1}[f]\right] (y)=\sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\left[ \int_{\mathbb{D}}f(z)\overline{W_{B}[\ell _{k}^{B}](z)}% d\eta _{B}(z)\right] \ell _{k}^{B}(y). \end{equation} We again apply $W_{B}$ to $\eqref{r5.9}:$ \begin{equation} \label{r5.10} W_{B}\left[ \wp _{R}^{B}\left[ W_{B}^{-1}[f]\right] \right] (w)=\sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\left[ \int_{\mathbb{D}}f(z)% \overline{W_{B}[\ell _{k}^{B}](z)}d\eta _{B}(z)\right] W_{B}[\ell _{k}^{B}](w). \end{equation} Since $W_{B}[\ell _{k}^{B}](w)=C_{k}^{B}(w),$ then we obtain \begin{equation} \label{r5.11} \widetilde{\wp }_{R}^{B}[f](w)=\int_{\mathbb{D}}\left[ \sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\overline{C_{k}^{B}(z)}C_{k}^{B}(w)\right] d\eta _{B}(z). \end{equation} Hence, the kernel integral function is given by \begin{equation} \label{r5.12} P_{R}^{B}(z,w)=\sum_{k=0}^{+\infty }\lambda _{k}^{B,R}\overline{C_{k}^{B}(z)}% C_{k}^{B}(w). \end{equation} To write this kernel in a closed form, we recall $\eqref{r34} $ and $\eqref{r20} $, then $\eqref{r5.12} $ becomes% \begin{equation} \label{r5.13} P_{R}^{B}(z,w)=(2B-1)\sum_{k=0}^{+\infty }\left[ \frac{1}{B(k+1,2B-1)}% \int_{0}^{R}\rho ^{k}(1-\rho )^{2B-2}d\rho \right] \frac{\Gamma (2B+k)}{% k!\Gamma (2B)}\bar{z}^{k}w^{k} \end{equation} \begin{equation} \label{r5.14} =(2B-1)^{2}\sum_{k=0}^{+\infty }\frac{\Gamma (2B+k)\Gamma (2B+k)}{\Gamma (2B)\Gamma (2B)}\frac{1}{k!}\frac{(\bar{z}w)^{k}}{k!}\left( \int_{0}^{R}t^{k}(1-t)^{2B-2}dt\right) \end{equation} \begin{equation} \label{r5.15} =(2B-1)^{2}\int_{0}^{R}(1-t)^{2B-2}\left( \sum_{k=0}^{+\infty }\frac{% (2B)_{k}(2B)_{k}}{(1)_{k}}\frac{(t\bar{z}w)^{k}}{k!}\right) dt \end{equation} The sum inside the integral can be presented as the Gauss hypergeometric function ${}_{2}F_{1}$ as \begin{equation} \label{r5.16} P_{R}^{B}(z,w)=(2B-1)^{2}\int_{0}^{R}(1-t)^{2B-2}{}_{2}F_{1}(2B,2B,1,t\bar{z}% w)dt. \end{equation} By setting $2B=\alpha $, $\bar{z}w=\omega $ and \begin{equation} \label{r5.17} I_{R}=\int_{0}^{R}{}_{2}F_{1}(\alpha ,\alpha ,1;\omega t)(1-t)^{\alpha -2}dt. \end{equation} By making use of the formula \cite[p.316]{PBM} : \begin{equation} \label{r5.18} \int_{0}^{y}x^{c-1}(y-x)^{\beta -1}(1-xz)^{-\tau }{}_{2}F_{1}(a,b,c;wx)dx=% \mathcal{B}(c,\beta )y^{c+\beta -1}(1-yz)^{-\tau }F_{3}(\tau ,a,\beta ,b,c+\beta ,\frac{yz}{yz-1};wy), \end{equation} $y,Re(c),Re(\beta )>0;|arg(1-wy)|,|arg(1-z)|<\pi $, for parameters $% y=R,x=t,c=1,\beta =1,z=1,\tau =2-2B,a=2B,b=2B$, the integral $\eqref{r5.17} $ takes the form \begin{equation} \label{r5.19} I_{R}=R(1-R)^{\alpha -2}F_{3}(2-\alpha ,\alpha ,1,\alpha ,2;\frac{R}{R-1}% ,\omega R) \end{equation} Next, we may apply the transformation \cite[p.450]{PBM}: \begin{equation} \label{r5.20} F_{3}\left( a,a^{\prime },b,b^{\prime };a+a^{\prime },w,z\right) =(1-z)^{-b^{\prime }}F_{1}\left( a,b,b^{\prime };a+a^{\prime };w,\frac{z}{z-1% }\right) \end{equation} to rewrite the $F_{3}$ hypergeometric function in $\eqref{r5.19} $ as \begin{equation} \label{r5.21} F_{3}\left( 2-\alpha ,\alpha ,1,\alpha ,\frac{R}{R-1},\omega R\right) =(1-\omega R)^{-\alpha }F_{1}\left( 2-\alpha ,1,\alpha ,2,\frac{R}{R-1},% \frac{R\omega }{R\omega -1}\right) . \end{equation}% Therefore, Eq.$\eqref{r5.19} $ reads \begin{equation} \label{r5.22} I_{R}=R(1-R)^{\alpha -2}(1-\omega R)^{-\alpha }F_{1}\left( 2-\alpha ,1,\alpha ,2,\frac{R}{R-1},\frac{R\omega }{R\omega -1}\right) . \end{equation} By applying the transformation \begin{equation} \label{r5.23} F_{1}\left( a,b_{1},b_{2},c;X;Y\right) =(1-X)^{-b_{1}}(1-Y)^{-b_{2}}F_{1}\left( c-a,b_{1},b_{2};c,\frac{X}{X-1},% \frac{Y}{Y-1}\right) \end{equation} we may reduce $I_{R}$ as \begin{equation} \label{r5.24} I_{R}=R(1-R)^{\alpha -1}F_{1}\left( \alpha ,1,\alpha ,2;R,R\omega \right) . \end{equation} Next, by using the symmetry relation \begin{equation} \label{r5.25} F_{1}\left( a,b,b^{\prime };c,z,u\right) =F_{1}\left( a,b^{\prime },b;c,u,z\right) \end{equation} together with the identity \cite[p.449]{PBM} \begin{equation} \label{r5.26} F_{1}\left( a,b,b^{\prime };c,u,z\right) =(1-u)^{c-a-b}(1-z)^{-b^{\prime }}F_{1}\left( c-a,c-b-b^{\prime },b^{\prime },c,u,\frac{u-z}{1-z}\right) \end{equation} enable us to rewrite $I_{R}$ as \begin{equation} \label{r5.27} I_{R}=\frac{R}{(1-R\omega )^{2B}}F_{1}\left( 2-2B,1-2B,2B,2,R,\frac{% R-R\omega }{1-R\omega }\right) \end{equation} Therefore, the kernel function $\eqref{r5.16} $ reads \begin{equation} \label{r5.28} P_{R}^{B}(z,w)=\frac{(2B-1)^{2}R}{(1-R\bar{z}w)^{2B}}F_{1}\left( 2-2B,1-2B,2B,2,R,\frac{R-R\bar{z}w}{1-R\bar{z}w}\right) . \end{equation} To check the limit of this kernel as $R\rightarrow 1$, we first observe that \begin{equation} \label{r5.29} F_{1}\left( 2-2B,1-2B,2B,2;R,\frac{R-R\bar{z}w}{1-R\bar{z}w}\right) \rightarrow F_{1}\left( 2-2B,1-2B,2B,2,1,1\right) \text{ as \ }R\rightarrow 1. \end{equation} By using the identity \cite[p.452]{PBM} \begin{equation} \label{r5.30} F_{1}\left( a,b,b^{\prime },c;Z;Z\right) ={}_{2}F_{1}\left( a,b+b^{\prime },c;Z\right) , \end{equation} the obtained expression limit in $\eqref{r5.29}$ reduces as \begin{equation} \label{r5.31} F_{1}\left( 2-2B,1-2B,2B,2B,2;1,1\right) ={}_{2}F_{1}\left( 2-2B,1,2;1\right) . \end{equation} Finally, we use the Gauss theorem \cite[p.489]{PBM}: \begin{equation} \label{r5.32} {}_{2}F_{1}(a,b,c;1)=\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)% }, \quad \rm{Re}(c-a-b)>0 \end{equation} for parameters $a=2-2B,b=1$ and $c=2$ to get that \begin{equation} \label{r5.33} {}_{2}F_{1}\left( 2-2B,1,2;1\right) =\frac{1}{2B-1}. \end{equation} This, leads to the limit \begin{equation} \label{r5.34} \lim_{R\rightarrow 1}P_{R}^{B}(z,w)=\frac{(2B-1)}{(1-\bar{z}w)^{2B}} \end{equation} This completes the proof. \smallskip We end this section by observing that $\eqref{r5.8} $ may provides us with a family of Hilbert spaces $\left( \text{RKHS}\right) $ indexed by the continuous parameter $R\in \left] 0,1\right[ ,$ where each one would have $% P_{R}^{B}(z,w)$ as its reproducing kernel for which the Eq. $\eqref{r5.12} 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