Orthogonal polynomials attached to coherent states for the symmetric Poschl-Teller oscillator

TL;DR

This paper introduces a new family of nonlinear coherent states linked to the symmetric Poschl-Teller oscillator, exploring their associated orthogonal polynomials and transformations, with implications for quantum systems like the infinite square well.

Contribution

It develops a novel class of nonlinear coherent states using generalized factorials and investigates their related orthogonal polynomials and Bargmann-type transforms.

Findings

Defined a new family of nonlinear coherent states for the Poschl-Teller potential.

Identified two sets of orthogonal polynomials associated with these states.

Extended results to the infinite square well potential.

Abstract

We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients of the canonical coherent states by a specific generalized factorial depending on a parameter gamma. These states are superposition of eigenstates of the Hamiltonian with a symmetric Poschl-Teller potential depending on a parameter nu > 1. The associated Bargmann-type transform is defined for equal parameters. Some results on the infinite square well potential are also derived. For some different values of gamma, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.