Some Noncommutative Matrix Algebras Arising in the Bispectral Problem

TL;DR

This paper extends the bispectral problem to include matrix coefficients and eigenvalues, exploring new classes of noncommutative matrix algebras with potential implications for integrable systems.

Contribution

It introduces a generalized framework for the bispectral problem involving matrix-valued differential operators and eigenfunctions, expanding previous scalar-based approaches.

Findings

Development of new noncommutative matrix algebras

Extension of bispectral problem to matrix eigenvalues

Examples illustrating the broader class of solutions

Abstract

I revisit the so called "bispectral problem" introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues, to be matrix valued too. In the last example we go beyond this and allow both eigenvalues to be matrix valued.