Corrigendum on the proof of completeness for exceptional Hermite polynomials

TL;DR

This paper addresses a gap in the proof of completeness for exceptional Hermite polynomials by providing a direct alternative proof utilizing the theory of trivial monodromy potentials.

Contribution

It offers a new, direct proof of the completeness of exceptional Hermite polynomials, correcting a previously identified gap in the original proof.

Findings

A corrected proof of completeness for exceptional Hermite polynomials

Application of trivial monodromy potential theory to polynomial completeness

Clarification of the mathematical foundation for exceptional orthogonal polynomials

Abstract

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio Dur\'an discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Gr\"unbaum and Oblomkov.