On the Differential Operators with Periodic Matrix Coefficients

TL;DR

This paper derives asymptotic formulas for eigenvalues and eigenfunctions of differential operators with periodic matrix coefficients, and identifies conditions for the spectrum to have finitely many gaps.

Contribution

It provides new asymptotic formulas and spectral gap conditions for differential operators with periodic matrix coefficients.

Findings

Asymptotic formulas for eigenvalues and eigenfunctions

Conditions for finite spectral gaps

Analysis of operators with quasiperiodic boundary conditions

Abstract

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.