An application of the theta operator in generalized hypergeometric coherent states formalism

TL;DR

This paper explores the use of the theta operator in the formalism of generalized hypergeometric coherent states, revealing new integral representations and applying these to quantum systems with linear spectra.

Contribution

It introduces new applications of the theta operator within GHG CSs, expanding their mathematical framework and potential physical applications.

Findings

Derived new integral representations of hypergeometric functions

Confirmed existing results like Laplace transforms of hypergeometric functions

Applied the formalism to quantum systems with linear energy spectra

Abstract

In this paper we examine one of the multiple applications of the theta operator xd/dx in quantum mechanics, namely, in the formalism of generalized hypergeometric coherent states (GHG CSs). These states are the most general coherent states, in the sense that from them, through particularization, all coherent states with physical meaning can be obtained. A series of new results were obtained and some already known ones were found / confirmed (the integral representations, as well as the Laplace transform of hypergeometric functions). To support the theoretical considerations presented above, we examined, as example, the quantum systems with a linear energy spectrum. The results obtained in this paper contribute to widening the area of applicability of the theta operator.