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The Riesz Basis Property of an Indefinite Sturm–Liouville Problem with Non-Separated Boundary Conditions

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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.

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Authors and Affiliations

  1. Department of Mathematics, Western Washington University, Bellingham, WA, 98225, USA

    Branko Ćurgus

  2. Baroper Schulstrasse 27 A, 44225, Dortmund, Deutschland

    Andreas Fleige

  3. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Aleksey Kostenko

Authors
  1. Branko Ćurgus
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  2. Andreas Fleige
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  3. Aleksey Kostenko
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Correspondence to Aleksey Kostenko.

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Ćurgus, B., Fleige, A. & Kostenko, A. The Riesz Basis Property of an Indefinite Sturm–Liouville Problem with Non-Separated Boundary Conditions. Integr. Equ. Oper. Theory 77, 533–557 (2013). https://doi.org/10.1007/s00020-013-2093-x

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