New tools for the study of Bochner differential operators

TL;DR

This paper introduces a new sequence associated with Bochner differential operators, explores its properties, and uses it to solve inverse spectral problems, advancing understanding of eigenvalues and eigenpolynomials in this context.

Contribution

It presents a novel sequence for analyzing Bochner differential operators and characterizes conditions for their eigenvalues and eigenpolynomials, including inverse problem solutions.

Findings

Defined a sequence $\{\delta_n^{(k)} \\}$ for Bochner operators

Proved properties of this sequence and applied it to bispectral problems

Established necessary and sufficient conditions for inverse spectral problems

Abstract

A sequence $\{\delta_n^{(k)}\}$ associated to a Bochner differential operator is introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence $\{\lambda_n\}$ of complex numbers and a sequence $\{P_n\}$ of polynomials with complex coefficients, $\deg{P_n}=n$, we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on $\{\delta_n^{(k)}\}$.