A new family of shape invariantly deformed Darboux-P\"oschl-Teller potentials with continuous \ell

TL;DR

This paper introduces a new family of shape invariant potentials related to exceptional Jacobi and Laguerre polynomials, extending the class of conditionally exactly solvable quantum potentials with continuous parameters.

Contribution

It develops a continuous ll version of shape invariant potentials linked to exceptional orthogonal polynomials, expanding the scope of exactly solvable models in quantum mechanics.

Findings

Introduces a new continuous ll shape invariant potential family.

Connects these potentials to exceptional Jacobi and Laguerre polynomials.

Shows reduction to known conditionally exactly solvable potentials.

Abstract

We present a new family of shape invariant potentials which could be called a ``continuous \ell version" of the potentials corresponding to the exceptional (X_{\ell}) J1 Jacobi polynomials constructed recently by the present authors. In a certain limit, it reduces to a continuous \ell family of shape invariant potentials related to the exceptional (X_{\ell}) L1 Laguerre polynomials. The latter was known as one example of the `conditionally exactly solvable potentials' on a half line.