On periodic boundary value problem for the Sturm-Liouville operator

TL;DR

This paper investigates the basis properties of root functions for the Sturm-Liouville operator with periodic or antiperiodic boundary conditions, showing that the Fourier coefficients of the potential determine the basis property.

Contribution

It provides new insights into how the Fourier coefficients of the potential influence the basis property of root functions in Sturm-Liouville problems.

Findings

Root functions may or may not form a basis depending on Fourier coefficients.

The basis property is sensitive to the potential's Fourier coefficients.

Conditions are identified under which the root functions form a basis.

Abstract

We consider the Sturm-Liouville operator Lu=u''-q(x)u with periodic or antiperiodic boundary conditions. It is shown that depending of Fourier coefficients of the potential q(x) the system of root functions may have or may not have the basis property.