Simplicity of singular spectrum in anderson-type hamiltonians

We study self-adjoint operators of the form H_ω = H ₀ + ∑ ω(n)(δ_n|·) δ_n, where the δ_n'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 H_ω, and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of H_ω is a.s. simple.