Uniform Convergence of the Spectral Expansion for a Differential Operator with Periodic Matrix Coefficients

TL;DR

This paper derives asymptotic formulas for eigenvalues and eigenfunctions of a differential operator with periodic matrix coefficients, establishing conditions for Riesz basis formation and uniform spectral expansion convergence.

Contribution

It provides new asymptotic formulas and conditions ensuring the root functions form a Riesz basis for operators with periodic matrix coefficients.

Findings

Eigenvalue and eigenfunction asymptotics derived

Conditions for Riesz basis formation established

Uniform convergence of spectral expansion proved

Abstract

In this paper, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and the quasiperiodic boundary conditions. Using these asymptotic formulas, we find conditions on the coefficients for which the root functions of this operator form a Riesz basis. Then we obtain the uniformly convergent spectral expansion of the differential operators with the periodic matrix coefficients