Spectral and dynamical properties of random models with nonlocal and singular interactions

We give a spectral and dynamical description of certain models of random Schrödinger operators on L²(ℝ^dd) for which a modified version of the fractional moment method of Aizenman and Molchanov [3] can be applied. One family of models includes Schrödinger operators with random nonlocal interactions constructed from multidimensional wavelet bases. The second family includes Schrödinger operators with random singular interactions randomly located on sublattices of ℤ^d, for d = 1, 2, 3. We prove that these models are amenable to Aizenman-Molchanov-type analysis of the Green's function, thereby eliminating the use of multiscale analysis. The basic technical result is an estimate on the expectation of fractional moments of the Green's function. Among our results, we prove a Wegner estimate, Hölder continuity of the integrated density of states, and spectral and Hilbert-Schmidt dynamical localization at negative energies.