Simplicity of singular spectrum in Anderson-type Hamiltonians

Abstract

We study self-adjoint operators of the form $H_{\omega }=H_0+\sum \omega (n)({\delta }_n|·){\delta }_n$, where the ${\delta }_n$'s are a family of orthonormal vectors and the $\omega (n)$'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 $H_{\omega }$, and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of $H_{\omega }$ is a.s. simple