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Zouhair Mouayn

Publications and source records attributed to Zouhair Mouayn.

At least 19 recordsLinked to original sources
arXiv2024

A phase space localization operator in negative binomial states

We are dealing with some spectral properties of a phase space localization operator PR 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 D and depending on a parameter 2B > 1. We derive a formula expressing PR 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 PR to a Bergman space AB(D) of analytic functions on D satisfying a growth condition depending on B and we explicitly give its integral kernel whose limit as R goes to 1 coincides with the reproducing kernel of AB(D). This leads to a natural generalization of this Hilbert space with respect to the parameter R.

math-ph↗
arXiv2022

A fractal uncertainty principle for Bergman spaces and analytic wavelets

Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time Fourier transform (STFT) with a Gaussian window, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces on the unit disk, where the Cantor set is defined as a union of annuli that are equidistributed in the hyperbolic measure.The result can be written in terms of analytic Cauchy wavelets. As in the case of the STFT considered by Knutsen, our result consists of a two-sided bound for the norm of a localization operator involving the fractal dimension log 2 / log 3 in the exponent. As in the STFT case and in Dyatlov fractal uncertainty principle, the (hyperbolic) measure of the dilated iterates of the Cantor set in the disk tends to infinity, while the corresponding norm of the localization operator tends to zero.

math-ph↗
arXiv2021

A q deformation of true-polyanalytic Bargmann transforms when q^{-1}> 1

We combine continuous $q^{-1}$-Hermite Askey polynomials with new $2D$ orthogonal polynomials introduced by Ismail and Zhang as $q$-analogs for complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter $m$. In the analytic case corresponding to $m=0$, we recover a known result on the Ar\"ık-Coon oscillator for $q'=q^{-1}>1$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform on the complex plane. The obtained result may be used to introduce a $q$-deformed Ginibre-type point process.

math-ph↗
arXiv2020

Coherent States of Systems with Pure Continuous Energy Spectra

While dealing with a Hamiltonian with continuous spectrum we use a tridiagonal method involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonality measure to show that the obtained coherent states can be recast in the Gazeau-Klauder approach. The Hamiltonian of the $\ell$-wave free particle is treated as an example to illustrate the method.

math-ph↗
arXiv2020

Polyanalytic Reproducing Kernels on the Quantized Annulus

While dealing with the constant-strength magnetic Laplacian on the annulus, we complete J. Peetre's work. In particular, the eigenspaces associated with its discrete spectrum are true-polyanalytic spaces with respect to the invariant Cauchy-Riemann operator, and we write down explicit formulas for their reproducing kernels. The latter are expressed by means of the fourth Jacobi theta function and of its logarithmic derivatives when the magnetic field strength is an integer. Under this quantization condition, we also derive the transformation rule satisfied by the reproducing kernel under the automorphism group of the annulus.

math-ph↗
arXiv2020

A Generalized Euler Probability Distribution

From a new class of q-deformed coherent states we introduce a generalization of the Euler probability distribution for which the main statistical parameters are obtained explicitly. As application, we discuss the corresponding photon counting statistics with respect to all parameters labeling the coherent states under consideration.

math-ph↗
arXiv2017

A generating function and formulae defining the first-associated Meixner-Pollaczek polynomials

While considering nonlinear coherent states with specific anti-holomorphic coefficients $\bar{z}^n/\sqrt{x_n!}$, we identify as first associated Meixner-Pollaczek polynomials the orthogonal polynomials arising from shift operators which are defined by the sequence $x_n=(n+1)^2$ . We give a formula defining these polynomials by writing down their generating function. This also leads to construct a Bargmann-type integral transform whose kernel is given in terms of a $Ψ_1$ Humbert's function.

math-ph↗
arXiv2016

Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator

We construct a new class of coherent states indexed by points z of the complex plane and depending on two positive parameters m and epsilon by replacing the coefficients of the canonical coherent states by polyanalytic functions. These states solve the identity of the states Hilbert space of the harmonic oscillator at the limit epsilon goes to zero and obey a thermal stability property. Their wavefunctions are obtained in a closed form and their associated Bargmann-type transform is also discussed.

math-ph↗
arXiv2016

Berezin transforms attached to Landau levels on the complex projective space CPn

In this paper, we construct coherent states for each generalized Bergman space on the n-dimensional complex projective space in order to apply a coherent states quantization method. Doing so allows to define the Berezin transform for these spaces. In particular, we provide a variational formula for this transform by means of the Fubini-Study Laplace operator which reduces when n = 1 and for the lowest spherical Landau level to the well-known formula previously given by Berezin himself.

math-ph↗
arXiv2016

Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels

To each hyperbolic Landau level of the Poincaré disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its decomposition as a perturbation of the negative binomial distribution by a finitely-supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we determine the Lévy-Kintchine decomposition its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-Lévy measure, we introduce a new infinitely-divisible distribution for which we derive the characteristic function.

math.PR↗
arXiv2016

Orthogonal polynomials attached to coherent states for the symmetric Poschl-Teller oscillator

We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients of the canonical coherent states by a specific generalized factorial depending on a parameter gamma. These states are superposition of eigenstates of the Hamiltonian with a symmetric Poschl-Teller potential depending on a parameter nu > 1. The associated Bargmann-type transform is defined for equal parameters. Some results on the infinite square well potential are also derived. For some different values of gamma, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.

math-ph↗
arXiv2015

Supersymmetry of the Morse oscillator

While dealing in [1] with the supersymmetry of a tridiagonal Hamiltonian H, we have proved that its partner Hamiltonian H(+) also have a tridiagonal matrix representation in the same basis and that the polynomials associated with the eigenstates expansion of H(+) are precisely the kernel polynomials of those associated with H. This formalism is here applied to the case of the Morse oscillator which may have a finite discrete energy spectrum in addition to a continuous one. This completes the treatment of tridiagonal Hamiltonians with pure continuous energy spectrum, a pure discrete one, or a spectrum of mixed discrete and continous parts.

math-ph↗
arXiv2015

Polyanalytic relativistic second Bargmann transforms

We construct coherent states through special superpositions of photon number states of the relativistic isotonic oscillator. In each superposition the coefficients are chosen to be L 2 eingenfunctions of a sigma weight Maass Laplacian on the Poincare disk, which are associated with discrete eigenvalues. For each nonzero m the associated coherent states transform constitutes the m true polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel Kampe de Feriet hypergeometric function. The obtained results could be used to extend the known semi classical analysis of quantum dynamics of the relativistic isotonic oscillator.

math-ph↗
arXiv2015

Analysis of generalized probability distributions associated with higher Landau levels

To a higher Landau Level corresponds a generalization of the Poisson distribution arising from generalized coherent states. In this paper, we write down the atomic decomposition of this probability measure and expressed its weights through 2F2 hypergeometric polynomials. Then, we prove that it is not infinitely divisible in opposite to the Poisson distribution corresponding to the lowest Landau level. We also derive the Levy-Kintchine representation of its characteristic function when the latter does not vanish and deduce that the representative measure is signed. By considering the total variation of the last measure, we obtain the characteristic function of a new infinitely divisible discrete probability distribution for which we compute also the weights.

math.PR↗
arXiv2015

New coherent states with Laguerre polynomials coefficients for the symmetric Poschl-Teller oscillator

We construct a new class of coherent states labeled by points z of the complex plane and depending on three numbers (gamma, nu) and epsilon positive by replacing the coefficients of the canonical coherent states by Laguerre polynomials. These states are superpositions of eigenstates of the symmetric Poschl-Teller oscillator and they solve the identity of the states Hilbert space at the limit epsilon goes to 0. Their wavefunctions are obtained in a closed form for a special case of parameters (gamma, nu). We discuss their associated coherent states transform which leads to an integral representation of Hankel type for Laguerre functions.

math-ph↗
arXiv2012

Spin coherent states with monopole harmonics on the Riemann sphere for the Kravchuk oscillator

We consider a class of generalized spin coherent states by choosing the labeling coefficients to be monopole harmonics.The latters are L2 eigenstates of the mth spherical Landau level on the Riemann sphere with m in Z+. We verify that the Klauder minimum properties for these states to be considered as coherent states are satisfied. We particularize them for the case of the Kravchuk oscillator and we obtain explicite expression for their wave functions.The associated coherent states transforms provide us with a Bargmann-type representation for the states of the oscillator Hilbert space. For the lowest level m = 0 indexing monopole harmonics, we identify the obtained coherent states to be those of Klauder-Perelomov type which were constructed in Ref.J. Math. Phys. 48, 112106 (2007)

math-ph↗
arXiv2011

Coherent States Quantization for Generalized Bargmann Spaces with Formulae for their Attached Berezin Transforms in Terms of the Laplacian on Cn

While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of such formulae is discussed.

math.CV↗