Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics

TL;DR

This paper explores the role of exceptional orthogonal polynomials in constructing new exactly solvable quantum potentials, highlighting their connection with rational extensions and supersymmetric quantum mechanics.

Contribution

It introduces a novel class of shape invariant potentials derived from exceptional orthogonal polynomials using higher-order SUSYQM techniques.

Findings

Exceptional orthogonal polynomials form complete orthogonal sets starting from degree ≥ 1.

Rational extensions of quantum potentials can be constructed using SUSYQM.

New shape invariant potentials are associated with EOP in quantum mechanics.

Abstract

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.