Peter D. Hislop, Werner Kirsch, M. Krishna
Spectral and Dynamical Properties of Random Models with Nonlocal and Singular Interactions
(147K, LaTex 2e)

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.