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A Class of Matrix Schrödinger Bispectral Operators

Brian D. Vasquez Campos

Abstract

We prove the bispectrality of some class of matrix Schrödinger operators with polynomial potentials that satisfy a second-order matrix autonomous differential equation. The physical equation is constructed using the formal theory of the Laurent series and after that obtaining local solutions using estimations in the Frobenius norm. Furthermore, the characterization of the algebra of polynomial eigenvalues in the spectral variable is given using some family of functions $\left\{ P_{k}\right\}_{k\in \mathbb{N}}$ with the remarkable property of satisfying a general version of the Leibniz rule.

A Class of Matrix Schrödinger Bispectral Operators

Abstract

We prove the bispectrality of some class of matrix Schrödinger operators with polynomial potentials that satisfy a second-order matrix autonomous differential equation. The physical equation is constructed using the formal theory of the Laurent series and after that obtaining local solutions using estimations in the Frobenius norm. Furthermore, the characterization of the algebra of polynomial eigenvalues in the spectral variable is given using some family of functions with the remarkable property of satisfying a general version of the Leibniz rule.
Paper Structure (8 sections, 20 theorems, 158 equations)

This paper contains 8 sections, 20 theorems, 158 equations.

Table of Contents

  1. Algebraic Morphisms Arising from the Matrix Equation Lg
  2. The Matrix Equation $V^{\prime \prime}(x)=V^{\prime}(x)V(x)$.
  3. Some Properties of the Sequence $\left\{ V_{k}(V_{0},V_{1},V_{2})\right\}_{k\in \mathbf{\mathbb{N}}}$
  4. Polynomial Solutions of the matrix equation $V"(x)=V'(x)V(x)$
  5. Bispectrality of the Matrix Schrödinger Bispectral Operators for polynomial potentials
  6. Illustrative Examples
  7. The Bispectral Algebra Associated to the Potential with Invertible Residue at $x=0$
  8. Examples of Polynomial Potentials of degree $n=1,2,3$

Key Result

Lemma 1

$L_{A}$ and $R_{A}$ commutes with $T_{k}$ for $k\geq 1$ if, and only if, $A_{12}=0\in M_{m\times (N-m)}(\mathbf{\mathbb{C}})$ and $A_{21}=0\in M_{(N-m)\times m}(\mathbf{\mathbb{C}})$.

Theorems & Definitions (44)

  • Definition 1
  • Lemma 1
  • Proof
  • Lemma 2
  • Proof
  • Remark 1
  • Theorem 1
  • Proof
  • Corollary 1
  • Proof
  • ...and 34 more