  We use recent results on the boundary behavior of Cauchy integrals to study
the Krein spectral shift of a rank one perturbation problem for self-adjoint
operators. As an application, we prove that all self-adjoint rank one
perturbations of a self-adjoint operator are pure point if and only if the
spectrum of the operator is countable. We also study pairs of pure point
operators unitarily equivalent up to a rank one perturbation and give various
examples of rank one perturbations of singular spectra.
