Opuscula Math. 36, no. 6 (2016), 717-734
http://dx.doi.org/10.7494/OpMath.2016.36.6.717

Opuscula Mathematica

# Eigenvalue estimates for operators with finitely many negative squares

Jussi Behrndt
Roland Möws
Carsten Trunk

Abstract. Let $$A$$ and $$B$$ be selfadjoint operators in a Krein space. Assume that the resolvent difference of $$A$$ and $$B$$ is of rank one and that the spectrum of $$A$$ consists in some interval $$I\subset\mathbb{R}$$ of isolated eigenvalues only. In the case that $$A$$ is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of $$B$$ in the interval $$I$$. The general results are applied to singular indefinite Sturm-Liouville problems.

Keywords: selfadjoint operator, Krein space, finitely many negative squares, eigenvalue estimate, indefinite Sturm-Liouville operator.

Mathematics Subject Classification: 47A55, 47B50.

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• Jussi Behrndt
• Technische Universität Graz, Institut für Numerische Mathematik, Steyrergasse 30, 8010 Graz, Austria
• Roland Möws
• Krossener Str. 17, D-10245 Berlin, Germany
• Carsten Trunk
• Technische Universität Ilmenau, Institut für Mathematik, Postfach 100565, D-98684 Ilmenau, Germany
• Communicated by A. Shkalikov.
• Revised: 2016-05-24.
• Accepted: 2016-05-24.
• Published online: 2016-10-29.