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Journal of Physics A Mathematical and Theoretical

Arik–Coon oscillator withq> 1 in the framework of unified (q; α, β, γ; ν)-deformation

I M Burban

Published Jun 29, 2010• 8 citations
Publisher

Abstract

In this paper we study the Arik–Coon oscillator with the main relation aa + − qa + a = 1, where q > 1, in the framework of the unified ( q ; α, β, γ; ν)-deformed oscillator algebra. This oscillator is connected with the Askey q −1 -Hermite polynomials. We construct a family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical moment problem, we prove the over-completeness of this family of coherent states.

DOI
10.1088/1751-8113/43/30/305204

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In this paper we study the Arik–Coon oscillator with the main relation aa + − qa + a = 1, where q > 1, in the framework of the unified ( q ; α, β, γ; ν)-deformed oscillator algebra. This oscillator is connected with the Askey q −1 -Hermite polynomials. We construct a family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical moment problem, we prove the over-completeness of this family of coherent states.
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The aim of this paper is to review our results on description of the multi-parameter deformed oscillators and their oscillator algebras. We define generalized (q;α,β,γ;ν)-deformed oscillator algebra and study its irreducible representations. The Arik-Coon oscillator with the main relation aa^+ - q a^+a = 1, where q > 1 is embedded in this framework. We find connection of this oscillator with the Askey (1/q)-Hermite polynomials. We construct family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of appropriate classical Stielties moment problem we prove the (over)completeness relation of these states.
The aim of this paper is the study of the generalized deformed quantum oscillators in the framework (q; α, β, γ; ν)-deformed of oscillator algebra. By selecting the special values of deformation parameters, we have separated a generalized deformed oscillator connected with generalized discrete Hermite II polynomials. By these means we have constructed Barut–Girardello type coherent states of this oscillator. We have found the conditions on the (q; α, β, γ; ν)-deformation parameters at which the free (q; α, β, γ; ν)-deformed oscillator approximate the usual anharmonic oscillator in the homogeneous Kerr medium.
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.Our construction leads to a new qdeformation of the m-true-polyanalytic Bargmann transform on the complex plane.In the analytic case m = 0, the obtained coherent states transform can be associated with the Arïk-Coon oscillator for q = q -1 > 1.These result may be used to introduce a q-deformed Ginibre-type point process.
A set of qqq-coherent states for the Rogers-Szegő oscillator
We discuss a model of a qqq-harmonic oscillator based on Rogers-Szegő functions. We combine these functions with a class of qqq-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter mmm. Our construction leads to a new qqq-deformation of the mmm-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 qqq-deformation of the Ginibre-mmm-type process on the complex plane.