Infinitely many shape invariant potentials and new orthogonal polynomials

TL;DR

This paper introduces infinitely many new exactly solvable quantum potentials derived from deformations of known potentials, along with new orthogonal polynomials, expanding the class of shape invariant models.

Contribution

It presents a systematic construction of infinitely many shape invariant potentials and their eigenfunctions expressed via novel orthogonal polynomials, extending previous specific cases.

Findings

Three sets of exactly solvable potentials are constructed.

Eigenfunctions are expressed in terms of new orthogonal polynomials.

The first members of these families include previously reported potentials.

Abstract

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic P\"oschl-Teller potentials in terms of their degree \ell polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (\ell=1,2,...) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and G\'omez-Ullate et al's are the first members of these infinitely many potentials.