Opuscula Math. 36, no. 6 (2016), 717-734

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.
  • Received: 2016-04-16.
  • Revised: 2016-05-24.
  • Accepted: 2016-05-24.
  • Published online: 2016-10-29.
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Cite this article as:
Jussi Behrndt, Roland Möws, Carsten Trunk, Eigenvalue estimates for operators with finitely many negative squares, Opuscula Math. 36, no. 6 (2016), 717-734, http://dx.doi.org/10.7494/OpMath.2016.36.6.717

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