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Journal of Physics A Mathematical and General

Generalized deformed oscillator corresponding to the modified Poschl-Teller energy spectrum

C. Daskaloyannis

Published Apr 21, 1992• 45 citations
Publisher

Abstract

A generalized deformed oscillator, with eigenvalues equal to the eigenvalues of the Schrodinger equation with the modified Poschl-Teller potential, is constructed. For special values of the potential depth the deformed oscillator algebra has a finite-dimensional irreducible representation. The polynomial representation and the associate deformed operations of integration and differentiation are studied. The results of this study are general and they can be applied directly to the case of the q-deformed oscillator, with q being a root of unity.

DOI
10.1088/0305-4470/25/8/036

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A generalized deformed oscillator, with eigenvalues equal to the eigenvalues of the Schrodinger equation with the modified Poschl-Teller potential, is constructed. For special values of the potential depth the deformed oscillator algebra has a finite-dimensional irreducible representation. The polynomial representation and the associate deformed operations of integration and differentiation are studied. The results of this study are general and they can be applied directly to the case of the q-deformed oscillator, with q being a root of unity.
Generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show that this generalized formalism is able (a) to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system, (b) to reproduce, as particular cases, results already known for shape-invariant systems (such as standard harmonic oscillator and Pöschl–Teller potentials as well as quantum deformed harmonic oscillator models) and (c) point to a formalism that provides a unified description of the different kind of coherent states for quantum systems, deformed or not deformed.
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.