On the self-adjoint differential operator with the periodic matrix coefficients

TL;DR

This paper analyzes the spectral properties of a self-adjoint differential operator with periodic matrix coefficients, establishing finiteness of spectral gaps, conditions for spectrum reality, and spectral band structure.

Contribution

It provides explicit estimates for the number of spectral gaps and conditions ensuring the spectrum lies on the real axis for operators with periodic matrix coefficients.

Findings

Number of spectral gaps is finite.

Explicit bounds on the number of gaps based on coefficients.

Conditions for the spectrum to be real on the entire axis.

Abstract

In this paper we consider the spectrum of the self-adjoint differential operator L generated by the differential expression of order n with the m by m periodic matrix coefficients, where n and m are respectively odd and even integers and n>1. We prove that the number of gaps in the spectrum of L is finite and find explicit estimation in term of coefficients for the number of the gaps. Moreover, we find a condition on the norms of the coefficients for which the spectrum is real axis. Besides we investigate the bands of the spectrum and prove that most of the real axis is overlapped by m bands.