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Integral Transforms and Special Functions

Research Article Orthogonal polynomials attached to coherent states for the symmetric Pöschl–Teller oscillator

Khalid Ahbli, Patrick Kayupe Kikodio, Zouhaı̈r Mouayn

Published Jul 20, 2016• 3 citations
Publisher

Abstract

We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients zn/n! of the canonical coherent states by a specific generalized factorial xnγ!, γ≥0. These states are superposition of eigenstates of the Hamiltonian with a symmetric Pöschl–Teller potential depending on a parameter ν>1. The associated Bargmann-type transform is defined for γ=ν. Some results on the infinite square well potential are also derived. For some different values of γ, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.

DOI
10.1080/10652469.2016.1210143

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We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients zn/n! of the canonical coherent states by a specific generalized factorial xnγ!, γ≥0. These states are superposition of eigenstates of the Hamiltonian with a symmetric Pöschl–Teller potential depending on a parameter ν>1. The associated Bargmann-type transform is defined for γ=ν. Some results on the infinite square well potential are also derived. For some different values of γ, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.
We construct new coherent states labeled by points z of the complex plane and depending on three parameters and by replacing the coefficients of the canonical coherent states by Laguerre polynomials with an order depending on γ . These coherent states are superpositions of eigenstates of the Hamiltonian with a symmetric Pöschl–Teller potential indexed by ν , which solve an -identity operator while the resolution of the identity of the states Hilbert space is acheived at the limit We obtain their wave functions in a closed form for a special case of parameters γ and ν . We also discuss their associated coherent states transform which leads to an integral representation of Hankel type for Laguerre functions.
Abstract We construct a new class of nonlinear coherent states for the isotonic oscillator by replacing the factorial of the coefficients z n / n ! zn/n!{z}^{n}/\sqrt{n!}zn/n!​ of the canonical coherent states by the factorial x n γ ! = x 1 γ . x 2 γ … x n γ xnγ!=x1γ.x2γ…xnγ{x}_{n}^{\gamma }!={x}_{1}^{\gamma }.{x}_{2}^{\gamma }\dots {x}_{n}^{\gamma }xnγ​!=x1γ​.x2γ​…xnγ​ with x 0 γ = 0 x0γ=0{x}_{0}^{\gamma }=0x0γ​=0 , where x n γ xnγ{x}_{n}^{\gamma }xnγ​ is a sequence of positive numbers and γ is a positive real parameter. This also leads to the construction of a Bargmann-type integral transform which will allow us to find some integral transforms for orthogonal polynomials. The statistics of our coherent states will also be considered by the calculus of one called Mandel parameter. The squeezing phenomenon was also discussed.
We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients zn/n!z^n/\sqrt{n!}zn/n!​ of the canonical coherent states by a specific generalized factorial xnγ,σ!x_n^{γ,σ}!xnγ,σ​! where parameters γγγ and σσσ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters γγγ and σσσ, we attache these states to the pseudo-harmonic oscillator depending on two parameters α,β>0α,β> 0α,β>0. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.
The Pöschl-Teller (PT) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications from quantum mechanics to diatomic molecules. For this potential, a polynomial su(1, 1) algebra has been constructed previously. So far, the coherent states (CSs) associated with this algebra have never appeared. In this paper, we construct the coherent states of the Barut-Girardello coherent states (BG-CSs) type for the PT potentials, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BG-CSs and Husimi’s and P-quasi distributions (for the mixed-thermal states) is also presented. Finally, the exponential form of the BG-CSs for the PT potential has been presented and enabled us to build Perelomov type CSs for the PT potential. We point out that the BG-CSs and the Perelomov type coherent states (PCSs) are related via Laplace transform.
This article is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Pöschl–Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry à la Barut-Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.
Coherent states in the harmonic oscillator may be defined in several equivalent ways. One definition describes coherent states as special states satisfying a minimum-uncertainty requirement in position and momentum spaces. This definition is generalized for other potentials according to a method developed by Nieto et al. [Phys. Rev. D 20, 1321 (1979)] and is herein applied to the P\"oschl-Teller potential. A classical limit based on the harmonic-oscillator coherent-state classical limit is then developed and applied to the P\"oschl-Teller minimum-uncertainty states. In this limit, classical behavior may be obtained from these quantum states. Together with a completeness argument for the generalized coherent states, this result provides insight into quantum classical correspondence through the statistical interpretation of quantum mechanics.