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