New SUSYQM coherent states for Poschl-Teller potentials: a detailed mathematical analysis

TL;DR

This paper provides a comprehensive mathematical analysis of new supersymmetric quantum mechanics-based coherent states for P"oschl-Teller potentials, including proofs, operator analysis, and extensions to broader potential classes.

Contribution

It develops detailed mathematical foundations for these coherent states, including resolution of unity, operator properties, and extensions to more general potentials.

Findings

Validated resolution of unity for the states

Analyzed operator domains and self-adjointness

Extended framework to larger class of potentials

Abstract

In a recent short note [Bergeron H, Gazeau J P, Siegl P and Youssef A 2010 EPL 92 60003], we have presented the nice properties of a new family of semi-classical states for P\"oschl-Teller potentials. These states are built from a supersymmetric quantum mechanics approach and the parameters of these "coherent" states are points in the classical phase space. In this article we develop all the mathematical aspects that have been left apart in the previous article (proof of the resolution of unity, detailed calculations of quantized version of classical observables and mathematical study of the resulting operators: problems of domains, self- adjointness or self-adjoint extensions). Some additional questions as asymptotic behavior are also studied. Moreover, the framework is extended to a larger class of P\"oschl-Teller potentials.