Exceptional orthogonal polynomials and the Darboux transformation

TL;DR

This paper extends the Darboux transformation to polynomial Sturm-Liouville problems, characterizing new exceptional orthogonal polynomials and explaining their shape-invariance through transformation properties.

Contribution

It introduces a novel application of Darboux transformations to polynomial problems and characterizes $X_m$ Laguerre polynomials via isospectral transformations.

Findings

Characterization of $X_m$ Laguerre polynomials using Darboux transformations

Shape-invariance of exceptional polynomials derived from transformation permutability

Extension of Darboux transformation framework to polynomial Sturm-Liouville problems

Abstract

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.