Rational extensions of the trigonometric Darboux-P\"oschl-Teller potential based on para-Jacobi polynomials

TL;DR

This paper introduces new rational extensions of the trigonometric Darboux-P"oschl-Teller potential using para-Jacobi polynomials, leading to novel eigenstates and orthogonal polynomials dependent on a continuous parameter.

Contribution

It develops a method to generate regular rational extensions of the potential based on para-Jacobi polynomials, expanding the class of exactly solvable models.

Findings

Constructed one-step rational extensions depending on an integer and a continuous parameter.

Derived new families of orthogonal polynomials associated with the extended potentials.

Established eigenstates linked to these new polynomial families.

Abstract

The possibility for the Jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them para-Jacobi polynomials. Such polynomials are used here to build seed functions of a Darboux-B\"acklund transformation for the trigonometric Darboux-P\"oschl-Teller potential. As a result, one-step regular rational extensions of the latter depending both on an integer index $n$ and on a continuously varying parameter $\lambda$ are constructed. For each $n$ value, the eigenstates of these extended potentials are associated with a novel family of $\lambda$-dependent polynomials, which are orthogonal on $\left] -1,1\right[ $.