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Journal of Physics A Mathematical and General

Generalized coherent states for associated hypergeometric-type functions

Mahouton Norbert Hounkonnou, Komi Sodoga

Published Aug 23, 2005• 13 citations
Publisher

Abstract

We introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric-type operators. This makes possible the definition of generalized coherent states for the associated eigenfunctions. These generalized coherent states satisfy the relevant properties, i.e., label continuity, overcompleteness, temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system.

DOI
10.1088/0305-4470/38/36/004

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We introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric-type operators. This makes possible the definition of generalized coherent states for the associated eigenfunctions. These generalized coherent states satisfy the relevant properties, i.e., label continuity, overcompleteness, temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system.
In continuation of our previous works J. Phys. A: Math. Gen. 35, 9355-9365 (2002), J. Phys. A: Math. Gen. 38, 7851 (2005) and Eur. Phys. J. D 72, 172 (2018), we investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal PPP-representation of the density operator.
We investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer’s G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal P-representation of the density operator.
The present paper starts from a previously deduced result, in which the ννν-function plays the role of the normalization function of generalized hypergeometric coherent states for quantum systems with a continuous spectrum. We have generalized this idea, obtaining a new function, the generalized ννν-function. By defining a discrete-continuous limit, we revealed a series of interesting properties that, in the last instance, allow the formulation and solution of new integrals involving the generalized ννν-functions which depend on both scalar arguments as well as those containing creation and annihilation operators, which generate the generalized hypergeometric coherent states. To our knowledge, the results obtained by us do not appear in the literature.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.