Inverse spectral analysis for a class of infinite band symmetric matrices

TL;DR

This paper addresses the inverse spectral problem for a specific class of infinite band symmetric matrices, providing a characterization of spectral functions and an algorithm for matrix recovery based on rational interpolation.

Contribution

It introduces a new characterization of spectral functions and an algorithm for reconstructing matrices from spectral data in the context of infinite band symmetric operators.

Findings

Characterization of spectral functions for the class of matrices

Necessary and sufficient conditions for spectral functions

An algorithm for matrix recovery from spectral functions

Abstract

This note deals with the direct and inverse spectral analysis for a class of infinite band symmetric matrices. This class corresponds to operators arising from difference quations with usual and inner boundary conditions. We give a characterization of the spectral functions for the operators and provide necessary and sufficient conditions for a matrix-valued function to be a spectral function of the operators. Additionally, we give an algorithm for recovering the matrix from the spectral function. The approach to the inverse problem is based on the rational interpolation theory.