On the Non-Self-adjoint Sturm-Liouville Operators in the Space of Vector-Functions

TL;DR

This paper derives asymptotic formulas for eigenvalues and eigenfunctions of non-self-adjoint Sturm-Liouville operators with matrix potentials, establishing conditions for the root functions to form a Riesz basis.

Contribution

It provides new asymptotic formulas for eigenvalues and eigenfunctions of non-self-adjoint matrix Sturm-Liouville operators and identifies conditions for Riesz basis formation.

Findings

Asymptotic formulas for eigenvalues and eigenfunctions derived

Condition for root functions to form a Riesz basis established

Results extend understanding of non-self-adjoint Sturm-Liouville operators

Abstract

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in space of vector-functions by the Sturm-Liouville equation with m by m matrix potential and the boundary conditions whose scalar case (m=1) are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.