Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form H_ω = H₀ + Σλ_ω(n) 〈δ_n, ·〉δ_n, where the δ_n's are a family of orthonormal vectors and the λ_ω(n)'s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n, m), if the cyclic subspaces corresponding to the vectors δ_n and δ_m are not completely orthogonal, then the restrictions of H_ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that "well behaved" absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases.