Noncommutative bispectral Darboux transformations

TL;DR

This paper proves a broad theorem on bispectrality of noncommutative Darboux transformations, unifying many known cases and classifying transformations for certain differential operators using advanced algebraic methods.

Contribution

It establishes a general bispectrality theorem for noncommutative Darboux transformations and classifies specific cases using quasideterminants and spectral theory.

Findings

Proves a general bispectrality theorem for noncommutative Darboux transformations.

Classifies bispectral Darboux transformations for rank one differential and Airy operators.

Unifies known bispectral Darboux transformations under a single theoretical framework.

Abstract

We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference operators with values in all noncommutative algebras. All known bispectral Darboux transformations are special cases of the theorem. Using the methods of quasideterminants and the spectral theory of matrix polynomials, we explicitly classify the set of bispectral Darboux transformations from rank one differential operators and Airy operators with values in matrix algebras. These sets generalize the classical Calogero-Moser spaces and Wilson's adelic Grassmannian.