Algebraic Heun operator and band-time limiting

TL;DR

This paper introduces algebraic Heun operators linked to bispectral pairs, providing new tools for constructing commuting operators in band-time limiting problems and extending finite case constructions.

Contribution

It generalizes the Heun operator concept to bispectral pairs and adapts finite case constructions to broader situations in band-time limiting.

Findings

Constructed algebraic Heun operators for bispectral pairs

Provided a simple method for commuting operators in band-time limiting

Extended finite case algebraic constructions to new scenarios

Abstract

We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting with the projection operators in problems of band-time limiting and it gives a way to adapt a construction first used by Perline in a purely finite setup to quite a few other situations. We also extend his algebraic construction to cover some purely finite cases where his proposal fails to give a useful commuting operator.