Spectral analysis of infinite Marchenko-Slavin matrices

TL;DR

This paper provides a complete spectral characterization of infinite Marchenko-Slavin matrices, offering an algorithm for reconstructing these matrices from their spectral functions, advancing understanding in spectral analysis of infinite matrices.

Contribution

It introduces a complete spectral characterization and a reconstruction algorithm for infinite Marchenko-Slavin matrices, extending recent spectral analysis techniques.

Findings

Spectral functions are fully characterized for the matrices.

An explicit algorithm for matrix reconstruction is provided.

The work extends spectral analysis methods to matrices with degenerations.

Abstract

This work tackles the problem of spectral characterization of a class of infinite matrices arising from the modelling of small oscillations in a system of interacting particles. The class of matrices under discussion corresponds to the infinite Marchenko-Slavin class. The spectral functions of these matrices are completely characterized and an algorithm is provided for the reconstruction of the matrix from its spectral function. The techniques used in this work are based on recent results for the spectral characterization of infinite band symmetric matrices with so-called degenerations.