1996/01/03 by Alexei Poltoratski, Poltoratski, Alexei G.
Mathematics · Physics and Astronomy · #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.math/9601206
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.