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The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with Non-Separated Boundary Conditions
Integral Equations and Operator Theory
  • Branko Ćurgus, Western Washington University
  • Andreas Fleige
  • Aleksey Kostenko
Document Type
Article
Publication Date
1-1-2013
Disciplines
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.

Citation Information
Branko Ćurgus, Andreas Fleige and Aleksey Kostenko. "The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with Non-Separated Boundary Conditions" Integral Equations and Operator Theory Vol. 77 Iss. 4 (2013) p. 533 - 557
Available at: http://works.bepress.com/branko_curgus/19/