Dynamical properties of random Schr\"odinger operators

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.