Spectral Theory of Exceptional Hermite Polynomials

TL;DR

This paper explores the spectral properties of exceptional Hermite polynomials, providing new proofs of their completeness and characterizing their gap sets through spectral theory techniques.

Contribution

It offers an alternative proof of completeness and characterizes gap sets for exceptional Hermite polynomials using spectral theory methods.

Findings

Proved completeness of exceptional Hermite polynomials.

Characterized gap sets for these polynomials.

Connected spectral theory with exceptional polynomial properties.

Abstract

In this paper we revisit exceptional Hermite polynomials from the point of view of spectral theory, following the work initiated by Lance Littlejohn. Adapting a result of Deift, we provide an alternative proof of the completeness of these polynomial families. In addition, using equivalence of Hermite Wronskians we characterize the possible gap sets for the class of exceptional Hermite polynomials.