Singular continuous spectrum under rank one perturbations and localization for random hamiltonians

We consider a selfadjoint operator, A , and a selfadjoint rank‐one projection, P , onto a vector, φ, which is cyclic for A . In terms of the spectral measure dμ A φ , we give necessary and sufficient conditions for A + λ P to have empty singular continuous spectrum or to have only point spectrum for a.e. λ. We apply these results to questions of localization in the one‐ and multi‐dimensional Anderson models.