Simplicity of singular spectrum in Anderson-type Hamiltonians

Abstract

We study self-adjoint operators of the form Hω=H0+∑ω(n)(δn|·)δn, where the δn's are a family of orthonormal vectors and the ω(n)'s are independent random variables with absolutely continuous probability distributions. We prove a general structural theorem that provides in this setting a natural decomposition of the Hilbert space as a direct sum of mutually orthogonal closed subspaces, which are a.s. invariant under Hω, and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of Hω is a.s. simple