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Non-real eigenvalues of singular indefinite Sturm-Liouville operators

Authors: Jussi Behrndt, Qutaibeh Katatbeh and Carsten Trunk
Journal: Proc. Amer. Math. Soc. 137 (2009), 3797-3806
MSC (2000): Primary 47A10; Secondary 47B50
Published electronically: July 10, 2009
MathSciNet review: 2529889
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Abstract: We study a Sturm-Liouville expression with indefinite weight of the form $ \mathrm{sgn}(-d^2/dx^2+V)$ on $ \mathbb{R}$ and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials $ V$ with a certain behaviour at $ \pm\infty$ we prove that there are no real eigenvalues and that the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to $ -d^2/dx^2+V$ in $ L^2(\mathbb{R})$. The general results are illustrated with examples.

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Additional Information

Jussi Behrndt
Affiliation: Department of Mathematics MA 6–4, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany

Qutaibeh Katatbeh
Affiliation: Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan

Carsten Trunk
Affiliation: Department of Mathematics, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany

Received by editor(s): November 14, 2008
Received by editor(s) in revised form: February 14, 2009, and February 23, 2009
Published electronically: July 10, 2009
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.