# On the spectrum of the differential operators of even order with periodic matrix coefficients

## TL;DR

This paper investigates the spectral properties of high-order self-adjoint differential operators with periodic matrix coefficients, identifying conditions that lead to a finite number of spectral gaps.

## Contribution

It provides new criteria for the coefficients of such operators to ensure a finite number of spectral gaps, advancing understanding of their spectral structure.

## Key findings

- Conditions for finite spectral gaps are established.
- Analysis of band and Bloch functions for the operator.
- Spectral spectrum structure characterized under specific coefficient conditions.

## Abstract

In this paper, we consider the band functions, Bloch functions and spectrum of the self-adjoint differential operator L with periodic matrix coefficients. Conditions are found for the coefficients under which the number of gaps in the spectrum of the operator L is finite