On the Basis Property of the Root Functions of Differential Operators with Matrix Coefficients

TL;DR

This paper derives asymptotic formulas for eigenvalues and eigenfunctions of differential operators with matrix coefficients and establishes conditions for their eigenfunctions to form a Riesz basis, advancing spectral theory in differential operators.

Contribution

It provides new asymptotic formulas and necessary and sufficient conditions for the Riesz basis property of eigenfunctions in matrix coefficient differential operators.

Findings

Asymptotic formulas for eigenvalues and eigenfunctions

Necessary and sufficient conditions for Riesz basis formation

Characterization of basis properties in differential operators

Abstract

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.