Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ignacio Bono Parisi, Inés Pacharoni

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.

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

Abstract

In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all weight matrices 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 of size , 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.

Paper Structure

This paper contains 11 sections, 16 theorems, 68 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical orthogonal polynomials
  4. Matrix-valued orthogonal polynomials and the algebra $\mathcal{D}(W)$
  5. Bispectral Darboux transformations
  6. Fundamental properties of the new weight matrices $W$
  7. Solutions of the Matrix Bochner Problem
  8. Explicit examples of size $2\times 2$
  9. Hermite weights of size 2
  10. Laguerre weights of size 2
  11. Jacobi weights of size 2

Key Result

Theorem 1.4

If $W = W_H$, $W_L$, or $W_J$ are the weight matrices introduced in Definition W-definition, then:

Theorems & Definitions (34)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 2.1
  • Proposition 2.2: GT07, Propositions 2.6 and 2.7
  • Proposition 2.3: GT07, Proposition 2.8
  • Proposition 2.4: CY18, Prop. 2.23
  • Definition 2.5
  • Proposition 3.1
  • ...and 24 more