Gazeau-Klauder type coherent states for hypergeometric type operators

TL;DR

This paper explores the mathematical properties of hypergeometric type operators, their eigenfunctions, and eigenvalues, leading to the construction of Gazeau-Klauder coherent states and revealing both known and new results.

Contribution

It provides a systematic analysis of shape invariant hypergeometric operators and introduces a unified approach to defining Gazeau-Klauder coherent states with new insights.

Findings

Parameter dependence of eigenfunctions and eigenvalues analyzed

Construction of Gazeau-Klauder coherent states from hypergeometric operators

Recovery of known results and discovery of new properties

Abstract

The hypergeometric type operators are shape invariant, and a factorization into a product of first order differential operators can be explicitly described in the general case. Some additional shape invariant operators depending on several parameters are defined in a natural way by starting from this general factorization. The mathematical properties of the eigenfunctions and eigenvalues of the operators thus obtained depend on the values of the involved parameters. We study the parameter dependence of orthogonality, square integrability and of the monotony of eigenvalue sequence. The obtained results allow us to define certain systems of Gazeau-Klauder coherent states and to describe some of their properties. Our systematic study recovers a number of well-known results in a natural unified way and also leads to new findings.