Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces

Let A and B be selfadjoint operators in a Krein space and assume that the resolvent difference of A and B is of rank one. In the case that A is nonnegative and I   is an open interval such that σ(A)∩Iσ(A)∩I consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of B in I. The general result is illustrated with eigenvalue estimates for singular indefinite Sturm–Liouville problems.