The Darboux process and a noncommutative bispectral problem

TL;DR

This paper explores the Darboux process within a noncommutative matrix setting, focusing on 2x2 matrices, and investigates its implications for a noncommutative bispectral problem, extending classical scalar results.

Contribution

It introduces a noncommutative version of the Darboux process applied to matrix rings, providing concrete examples and insights into the bispectral problem in this setting.

Findings

Extension of Darboux process to 2x2 matrices

Identification of bispectral properties in noncommutative context

Concrete matrix examples illustrating theoretical concepts

Abstract

The Darboux process, also known by many other names, played a very important role in some extremely enjoyable joint work that Hans and I did 25 years ago. I revisit a version of this problem in a case when scalars are replaced by matrices, i.e., elements of a non-commutative ring. Many of the issues studied here can be pushed to the case of a ring with identity, but my emphasis is on very concrete examples involving $2 \times 2$ matrices.