Jean-Marie Barbaroux, Werner Fischer, Peter M\"uller
Dynamical properties of random Schr\"odinger operators
(335K, Postscript)

ABSTRACT.   We study dynamical properties of random Schr\"odinger operators 
 $H^{(\omega)}$ defined on the Hilbert space $\ell^2(\bbZ^d)$ or 
 $L^2(\bbR^d)$. Building on results from existing multi-scale 
 analyses, we give sufficient conditions on $H^{(\omega)}$ to obtain 
 the vanishing of the diffusion exponent 
 $$ 
 \sigma_{\rm diff}^+ := \limsup_{T\rightarrow\infty } \frac{\log 
 \bbE \left(\la\la\vert X 
 \vert^2\ra\ra_{T,f_I(H^{(\omega)})\psi}\right) }{\log T}=0. 
 $$ 
Here $\bbE$ is the expectation over randomness, $f_{I}$ is any 
smooth characteristic function of a bounded energy-interval $I$ and 
$\psi$ is a state vector in the domain of $H^{(\omega)}$ with 
compact spatial support. The quantity $\la\la |X|^2 
\ra\ra_{T,\varphi}$ denotes the Cesaro mean up to time $T$ of the 
second moment of position $\la |X|^2\ra_{t,\varphi}$ at times $0\le 
t\le T$ of an initial state vector $\varphi$. 
If the Hilbert space is $\ell^2(\bbZ^d)$, the method of proof can be 
strengthened to yield dynamical localization. 
Under weaker assumptions, we also prove a theorem on the absence of 
diffusion. The results are applied to a randomly perturbed periodic 
Schr\"odinger operator on $L^2(\bbR^d)$, to a simple Anderson-type 
model on the lattice and to a model with a correlated random 
potential in continuous space.