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

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[ $.