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Z. Mouayn

Publications and source records attributed to Z. Mouayn.

6 recordsLinked to original sources
arXiv2022

Heat coefficients for magnetic Laplacians on the complex projective space $\mathbf{P}^{n}(\mathbb{C})$

Denoting by $Δ_ν$ the Fubini-Study Laplacian perturbed by a uniform magnetic field strength proportional to $ν$, this operator has a discrete spectrum consisting on eigenvalues $β_m, \ m\in\mathbb{Z}_+$, when acting on bounded functions of the complex projective $n$-space. For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $Δ_ν$. Using a suitable polynomial decomposition of the multiplicity of each $β_m$, we write down a trace formula for the heat operator associated with $Δ_ν$ in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $t\searrow 0^+$ by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $Δ_ν$.

math-ph↗
arXiv2021

Husmi Q-functions attached to hyperbolic Landau levels

We are concerned with a phase-space probability distribution which is known as Husimi $Q$-function of a density operator with respect to a set of coherent states $\vert\widetildeκ_{z,B,R,m}\rangle$ attached to an $m$th hyperbolic Landau level and labeled by points $z$ of an open disk of radius $R$, where $B>0$ is proportional to a magnetic field strength. For a density operator representing a projector on a Fock state $\left\vert j\right\rangle$ we obtain the $Q_{j}$ distribution and discuss some of its basic properties such as its characteristic function and its main statistical parameters. We achieve the same program for the thermal density operator (mixed states) of the isotonic oscillator for which we establish a lower bound for the associated thermodynamical potential. We recover most of the results of the Euclidean setting (flat case) as the parameter $R$ goes to infinity by making appeal to asymptotic formulas involving orthogonal polynomials and special functions. As a tool, we establish a summation formula for the special Kampé de Fériet function $\digamma _{2:0:0}^{1:2:2}$.

math-ph↗
arXiv2018

Wehrl entropies and Euclidean Landau levels

We are concerned with an information-theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase-space probability $Q^{(m)}_{\hatρ}=\left\langle z,m|\hatρ|z,m\right\rangle $ which is known as Husimi function, where $\hatρ$ is a density operator and $% \left|z,m\right\rangle $ are coherent states attached to an Euclidean $m$th Landau level. We obtain the Husimi function $Q^{(m)}_β$ of the thermal density operator $\hatρ_β$ of the harmonic oscillator, which leads by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of $Q^{(m)}_β$ such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions $Q^{(m)}_β,\ m=0,1,2,...$. For $m\geq1$, we establish an exact expression for the Wehrl entropy of the density operator $% \hatρ_β$ and we discuss the behavior of this entropy with respect to the temperature parameter $T=1/β$

math-ph↗
arXiv2015

Discrete coherent states for higher Landau levels

We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2, R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.

math-ph↗
arXiv2012

Coherent states quantization of generalized bergman spaces on the unit ball of cn with a new formula for their associated berezin transforms

While dealing with a class of generalized Bergman spaces on the unit ball, 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, a new formula representing these transforms a functions of the Laplace-Beltrami operator is established in terms ofWilson polynomials by using the Fourier-Helgason transform.

math.FA↗