Supersymmetry and Shape Invariance of exceptional orthogonal polynomials

TL;DR

This paper explores the supersymmetric quantum mechanics framework applied to exceptional Laguerre and Jacobi orthogonal polynomials, revealing shape invariance as key to their solvability.

Contribution

It demonstrates the shape invariance property of exceptional orthogonal polynomials within SUSYQM, linking their differential equations to supersymmetric Hamiltonians.

Findings

Shape invariance underpins the solvability of exceptional orthogonal polynomials.

Differential equations for EOPs can be expressed as eigenvalue problems similar to Schrödinger equations.

The SUSYQM framework provides a new perspective on the structure of exceptional orthogonal polynomials.

Abstract

We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schr\"odinger equation to define "Hamiltonians" enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the solubility of the differential equations associated with these polynomials.