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Darboux equivalence for matrix-valued orthogonal polynomials

Ignacio Bono Parisi, Inés Pacharoni, Ignacio Zurrián

TL;DR

The work develops necessary and sufficient criteria for when matrix-valued orthogonal polynomials admit Darboux transformations, and furnishes explicit construction methods. It analyzes the differential-operator algebra \\mathcal{D}(W) and related modules to determine Darboux-equivalence to diagonal matrices of classical polynomials and to identify Darboux-irreducible MVOPs. The authors extend the theory to general-size MVOPs, providing multi-block criteria and dual formulations, and illustrate with concrete examples including a 3x3 scalar-reducible case, a non-scalar counterexample, and a 3x3 irreducible case with partial reducibility to a 2x2 direct sum. These results offer practical tools for verifying Darboux relations and constructing explicit transformations, enriching the understanding of MVOPs beyond classical scalar cases.

Abstract

In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see when and how any given sequence of polynomials is Darboux related to a diagonal matrix of classic orthogonal polynomials. We also explore the notion of Darboux-irreducibility and study some sequences that are not a Darboux transformation of classical orthogonal polynomials.

Darboux equivalence for matrix-valued orthogonal polynomials

TL;DR

The work develops necessary and sufficient criteria for when matrix-valued orthogonal polynomials admit Darboux transformations, and furnishes explicit construction methods. It analyzes the differential-operator algebra \\mathcal{D}(W) and related modules to determine Darboux-equivalence to diagonal matrices of classical polynomials and to identify Darboux-irreducible MVOPs. The authors extend the theory to general-size MVOPs, providing multi-block criteria and dual formulations, and illustrate with concrete examples including a 3x3 scalar-reducible case, a non-scalar counterexample, and a 3x3 irreducible case with partial reducibility to a 2x2 direct sum. These results offer practical tools for verifying Darboux relations and constructing explicit transformations, enriching the understanding of MVOPs beyond classical scalar cases.

Abstract

In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see when and how any given sequence of polynomials is Darboux related to a diagonal matrix of classic orthogonal polynomials. We also explore the notion of Darboux-irreducibility and study some sequences that are not a Darboux transformation of classical orthogonal polynomials.
Paper Structure (9 sections, 10 theorems, 73 equations)

This paper contains 9 sections, 10 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Matrix valued orthogonal polynomials and the algebra $\mathcal{D}(W)$
  3. The Darboux transformation
  4. Darboux transformation of classical polynomials
  5. Example 1. Darboux transformation of a direct sum of classical scalar weights
  6. Example 2. A case that is not a Darboux transformation of scalar classical polynomials
  7. Darboux transformation of matrix-valued polynomials
  8. Example 3. Darboux-reducible to a direct sum of $2\times 2$ orthogonal polynomials.
  9. Example 4. An example of size $3$ that is not Darboux-reducible to scalars and $\mathcal{D}(W,w)\neq0$

Key Result

Theorem 3.3

If there exists a differential operator $\mathcal{V}\in \operatorname{Mat}_{N}(\Omega[x])$ such that with $A_{n} \in \operatorname{Mat}_{N}(\mathbb{C})$ invertible for all but finitely many $n$, then $\tilde{P}$ is a Darboux transformation of $P$. The converse is also true.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Remark
  • proof : Proof of Theorem \ref{['darboux-thm']}
  • Corollary 3.5
  • Definition 3.6
  • Proposition 3.7
  • ...and 15 more