On the Basis Property of the Root Functions of Sturm-Liouville Operators with General Regular Boundary Conditions

TL;DR

This paper derives asymptotic formulas for eigenvalues and eigenfunctions of Sturm-Liouville operators with general boundary conditions, identifying conditions where root functions fail to form a Riesz basis.

Contribution

It provides new asymptotic formulas and conditions that determine when the root functions of these operators do not constitute a Riesz basis.

Findings

Eigenvalues and eigenfunctions asymptotics derived

Sufficient conditions for root functions not forming a Riesz basis identified

Potential q influences basis properties of root functions

Abstract

We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with general 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.