Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

TL;DR

This paper characterizes the spectral data of vector-valued Sturm-Liouville operators on the unit interval, providing a complete description of Weyl-Titchmarsh functions for self-adjoint matrix potentials with distinct averaged eigenvalues.

Contribution

It offers a complete characterization of spectral data for matrix-valued Sturm-Liouville operators with distinct averaged eigenvalues, extending the understanding of Weyl-Titchmarsh functions.

Findings

Spectral data characterized for operators with distinct averaged eigenvalues.

Complete description of Weyl-Titchmarsh functions for matrix potentials.

Results applicable to self-adjoint operators with square-integrable potentials.

Abstract

The matrix-valued Weyl-Titchmarsh functions $M(\lambda)$ of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of $M(\lambda)$) and the residues of $M(\lambda)$ is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, $N\times N$ Weyl-Titchmarsh functions) corresponding to $N\times N$ self-adjoint square-integrable matrix-valued potentials is given, if all $N$ eigenvalues of the averaged potential are distinct.