The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions

TL;DR

This paper investigates conditions under which the eigenfunctions of an indefinite Sturm-Liouville problem form a Riesz basis, extending known results to non-separated boundary conditions and relating the basis property to boundary behavior of the weight function.

Contribution

It provides new necessary and sufficient conditions for the Riesz basis property in indefinite Sturm-Liouville problems with general boundary conditions, including non-separated cases.

Findings

Characterization of Riesz basis property for separated boundary conditions based on local behavior of r near turning points.

Extension of the characterization to certain non-separated boundary conditions requiring additional local conditions.

Connection between the Riesz basis property for periodic boundary conditions and a HELP-type inequality.

Abstract

We consider a regular indefinite Sturm-Liouville eigenvalue problem \{$-f" + q f = \lambda r f$} on $[a,b]$ subject to general self-adjoint boundary conditions and with a weight function $r$ which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions $r$ for which the Riesz basis property can be completely characterized in terms of the local behavior of $r$ in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of $r$ in a neighborhood of the turning points, local conditions on $r$ near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.