Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations

TL;DR

This paper derives four families of shape invariant Hamiltonians linked to exceptional Laguerre and Jacobi polynomials using Darboux-Crum transformations, expanding understanding of their properties and connections to known potentials.

Contribution

It provides a comprehensive derivation of shape invariant Hamiltonians for exceptional polynomials and explores their broader mathematical implications.

Findings

Connected shape invariant Hamiltonians to classical potentials

Expanded the method to include all families of exceptional polynomials

Discussed implications for the generalized Bochner problem and bispectrality

Abstract

Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-P\"oschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.