Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials

TL;DR

This paper introduces a systematic prepotential method to construct exactly solvable quantum systems related to exceptional Laguerre and Jacobi polynomials, bypassing traditional techniques like shape invariance.

Contribution

It develops a unified framework to derive potentials, eigenfunctions, and eigenvalues without assuming a pre-existing shape invariance or Darboux transformations.

Findings

All related quantum systems can be constructed systematically.

Exceptional polynomials are expressed as bilinear combinations.

The method simplifies the derivation of solvable potentials.

Abstract

We show how all the quantal systems related to the exceptional Laguerre and Jacobi polynomials can be constructed in a direct and systematic way, without the need of shape invariance and Darboux-Crum transformation. Furthermore, the prepotential need not be assumed a priori. The prepotential, the deforming function, the potential, the eigenfunctions and eigenvalues are all derived within the same framework. The exceptional polynomials are expressible as a bilinear combination of a deformation function and its derivative.