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Zouhaïr Mouayn

Publications and source records attributed to Zouhaïr Mouayn.

15 recordsLinked to original sources
arXiv2025

Zernike polynomials from the tridiagonalization of the radial harmonic oscillator in displaced Fock states

We revisit the J-matrix method for the one dimensional radial harmonic oscillator (RHO) and construct its tridiagonal matrix representation within an orthonormal basis phi(z)n of L2 (R+);parametrized by a fixed z in the complex unit disc D and n = 0,1,2,.... Remarkably, for fixed n,and varying z in D, the system phi(z)n forms a family of Perelomov-type coherent states associated with the RHO. For each fixed n, the expansion of phi(z)n over the basis (fs) of eigenfunctions of the RHO yields coefficients cn,s(z; z) precisely given by two-dimensional complex Zernike polynomials. The key insight is that the algebraic tridiagonal structure of RHO contains the complete information about the bound state solutions of the two-dimensional Schrödinger operator describing a charged particle in a magnetic field (of strength proportional to B > 1/2) on the Poincaré disc D.

math-ph↗
arXiv2025

Polyanalytic Hermite polynomials associated with the elliptic Ginibre model

Motivated by the connection between the eigenvalues of the complex Ginibre matrix model and the magnetic Laplacian in the complex plane, we derive analogues of the complex Hermite polynomials for the elliptic Ginibre model. To this end, we appeal to squeezed creation and annihilation operators arising from the Bogoliubov transformation of creation and annihilation operators on the Bargmann-Fock space. The obtained polynomials are then expressed as linear combinations of products of Hermite polynomials and share the same orthogonality relation with holomorphic Hermite polynomials. Moreover, this expression allows to identify them with the 2D-Hermite polynomials associated to a unimodular complex symmetric 2x2 matrix. Afterwards, we derive, for any Landau level, a closed formula for the kernel of the isometry mapping the basis of (rescaled) holomorphic Hermite polynomials to the corresponding complex Hermite polynomials. This kernel is also interpreted in terms of the two-photon coherent states and the metaplectic representation of the SU(1,1) group.

math-ph↗
arXiv2024

Generalized coherent states for the harmonic oscillator by the J-matrix method with an extension to the Morse potential

While dealing with the J-Matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L2(R); we rederive a set of generalized coherent states (GCS) of Perelomov type labeled by points z of the complex plane C and depending on a positive integer number m The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian associated with the mth Landau level. We extend this procedure to the Morse oscillator by constructing a new set of GCS of Glauber type.

quant-ph↗
arXiv2023

Mean and variance of the cardinality of particles in polyanalytic Ginibre processes via a quantization method

We discuss the mean and variance of the number \textquotedblleft point-particles\textquotedblright\ $\sharp _{D_{R}}$\ inside a disk $D_{R}$ centered at the origin of the complex plane $\mathbb{C}$ and of radius $R>0$ with respect to a Ginibre-type (polyanalytic) process of index $m\in \mathbb{Z}_{+}$ by quantizing the phase space $\mathbb{C} $ via a set of generalized coherent states $\left\vert z,m\right\rangle $ of the harmonic oscillator on $L^{2}\left(\mathbb{R}\right) $. By this procedure, the spectrum of the quantum observable representing the indicator function $χ_{D_{R}}$ of $ D_{R}$ (viewed as a classical observable) allows to compute the mean value of $\sharp _{D_{R}}$. The variance of $\sharp _{D_{R}}$ is obtained as a special eigenvalue of a quantum observable involving to the auto-convolution of $χ_{D_{R}}.$ By adopting a coherent states quantization approach, we seek to identify classical observables on $\mathbb{C},$ whose quantum counterparts may encode the first cumulants of $\sharp _{D_{R}}$ through spectral properties.

math-ph↗
arXiv2022

$q$-deformed coherent states associated with the sequence $x_n^{q,α}=(1+αq^{n-1})[n]_q$

We introduce new generalized $q$-deformed coherent states ($q$-CS) by replacing the $q$-factorial of $[n]_q!$ in the series expansion of the classical $q$-CS by the generalized factorial $x_n^{q,α}!$ where $x_n^{q,α}=(1+αq^{n-1})[n]_q$. We use the shifted operators method based on the sequence $x_n^{q,α}$ to obtain a realization in terms of Al-Salam-Chihara polynomials for the basis vectors of the Fock space carrying the constructed $q$-CS. These new states interpolate between the $q$-CS of Arik-Coon type ($α=0$, $0<q<1$) and a set of coherent states of Barut-Girardello type for the Meixner-Pollaczek oscillator ($α\neq 0$, $q\to 1$). We also discus their associated Bargmann type transforms.

math-ph↗
arXiv2021

A set of $q$-coherent states for the Rogers-Szegő oscillator

We discuss a model of a $q$-harmonic oscillator based on Rogers-Szegő functions. We combine these functions with a class of $q$-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter $m$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform whose range defines a generalization of the Arik-Coon space. We also give an explicit formula for the reproducing kernel of this space. The obtained results may be exploited to define a $q$-deformation of the Ginibre-$m$-type process on the complex plane.

math-ph↗
arXiv2021

Coherent states associated with tridiagonal Hamiltonians

It has been shown that a positive semi-definite Hamiltonian H, that has a tridiagonal matrix representation in a given basis, can be represented in the form H = A†A, where A is a forward shift operator playing the role of an annihilation operator. Such representation endows H with rich supersymmetric properties yielding results analogous to those obtained by studying the Hamiltonian as a differential operator. Here, we study the coherent states which we define as being the eigenstates of the operator A. We explicitly find the expansion coefficients of these states in the given basis. We further identify a complete set of special coherent states which themselves can be used as basis. In terms of these special coherent states, we show that a general coherent state has the expansion form of a Lagrange interpolation scheme. As application of the developed formalism, we work out examples of systems having pure discrete, pure continuous, or mixed energy spectrum.

math-ph↗
arXiv2018

Nonlinear coherent states associated with a measure on the positive real half line

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated coherent states transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this construction with the example of the measure $r^βe^{-r}dr$, which leads to a new generalization of the true-polyanalytic Bargmann transform.

math-ph↗
arXiv2018

Properties of shape-invariant tridiagonal Hamiltonians

It has been established that a positive semi-definite Hamiltonian,$H$, that has a tridiagonal matrix representation in a basis set, allows a definition of forward (and backward) shift operators that can be used to define the matrix representation of the supersymmetric partner Hamiltonian $H^{\left( +\right) \text{\ }}$ in the same basis. \ We show that if, additionally, the Hamiltonian has a shape invariant property, the matrix elements of the Hamiltonian are related in a such a way that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of super-symmetric partner Hamiltonians. Additionally, we derive the coherent states associated with this type of Hamiltonians and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also has tridiagonal matrix representation.

math-ph↗
arXiv2012

Husimi's Q-function isotonic oscillator in a generalized negative binomial states representation

While considering a class of generalized negative binomial states, we verify that the basic minimum properties for these states to be considered as coherent states are satisfied. We particularize them for the case of the Hamiltonian of the isotonic oscillator and we determine the corresponding Husimi's Q-function. This function may be used to determine a lower bound for the thermodynamical potential of the Hamiltonian by applying a Berezin-Lieb inequality

math-ph↗