Singular solutions of the matrix Bochner problem: the $N$-dimensional cases

TL;DR

This paper constructs new singular solutions to the matrix Bochner problem, which classify certain matrix-valued orthogonal polynomials that are not obtainable via known Darboux transformations, expanding the understanding of such solutions.

Contribution

It introduces three families of singular weight matrices related to classical weights, demonstrating solutions beyond the Darboux transformation framework.

Findings

Constructed weight matrices for Hermite, Laguerre, and Jacobi cases.

Identified solutions not obtainable by Darboux transformations.

Extended the classification of matrix orthogonal polynomials.

Abstract

In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all $N \times N$ weight matrices $W(x)$ such that the associated orthogonal polynomials are eigenfunctions of a second-order differential operator. In [4], Casper and Yakimov made an important breakthrough in this area, proving that, under certain hypotheses, every solution to this problem can be obtained as a bispectral Darboux transformation of a direct sum of classical scalar weights. In the present paper, we construct three families of weight matrices $W(x)$ of size $N \times N$, associated with Hermite, Laguerre, and Jacobi weights, which can be considered 'singular' solutions to the Matrix Bochner Problem because they cannot be obtained as a Darboux transformation of classical scalar weights.