On the Basis Property of the Root Functions of Some Class of Non-self-adjoint Sturm--Liouville Operators

TL;DR

This paper derives asymptotic formulas for eigenvalues and eigenfunctions of certain non-self-adjoint Sturm-Liouville operators and identifies conditions under which their root functions do not form a Riesz basis.

Contribution

It provides new asymptotic formulas and criteria for the basis property of root functions in non-self-adjoint Sturm-Liouville problems.

Findings

Eigenvalues and eigenfunctions asymptotics derived

Sufficient conditions for root functions not forming a Riesz basis established

Potential q influences basis property of root functions

Abstract

We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis.