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

Eigenvalue estimates for operators with finitely many negative squares

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.