02-386 Peter D. Hislop, Werner Kirsch, M. Krishna

Spectral and Dynamical Properties of Random Models with Nonlocal and Singular Interactions (147K, LaTex 2e) Sep 17, 02

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Abstract. We give a spectral and dynamical description of certain models of random Schr\"odinger operators on $L^2 ( \R^d)$ for which a modified version of the small moment method of Aizenman and Molchanov \cite{[AizenmanMolchanov]} can be applied. One family of models includes includes \Schr\ operators with random, nonlocal interactions constructed from a wavelet basis. The second family includes \Schr\ 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 small moments of the Green's function. Among our results, we prove a good Wegner estimate and the H\"older continuity of the integrated density of states, and spectral and dynamical localization at negative energies.

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