Spectral properties of matrix-valued discrete Dirac system

TL;DR

This paper analyzes the spectral properties of a matrix-valued discrete Dirac system, establishing the nature of its spectrum and eigenvalues through analytical and asymptotic methods.

Contribution

It introduces a polynomial-type Jost solution for the system and proves the continuous spectrum spans [-2,2], with finitely many simple real eigenvalues.

Findings

The continuous spectrum fills the segment [-2,2].

The system has finitely many simple real eigenvalues.

A polynomial-type Jost solution is constructed and analyzed.

Abstract

In this paper, we find a polynomial-type Jost solution of a self-adjoint matrix-valued discrete Dirac system. Then we investigate analytical properties and asymptotic behavior of this Jost solution. Using the Weyl compact perturbation theorem, we prove that matrix-valued discrete Dirac system has continuous spectrum filling the segment $[-2,2].$ Finally, we examine the properties of the eigenvalues of this Dirac system and we prove that it has a finite number of simple real eigenvalues.