Document Type

Article

Publication Date

2013

Keywords

Indefinite Sturm-Liouville problem, Riesz basis, Regular critical point, HELP inequality

Abstract

We consider a regular indefinite Sturm–Liouville eigenvalue problem −f′′ + q f = λ 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.

Publication Title

Integral Equations and Operator Theory

Volume

77

Issue

4

First Page

533

Last Page

557

Required Publisher's Statement

© Springer International Publishing AG

This is the author's version of this article. The published version can be found at the link below.

http://link.springer.com/article/10.1007%2Fs00020-013-2093-x

Subjects - Topical (LCSH)

Sturm-Liouville equation--Numerical solutions; Riesz spaces; Inequalities (Mathematics)

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

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Mathematics Commons

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