Eigenvalue estimates for operators with finitely many negative squares

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 of 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.