Dynamical properties of random Schrödinger operators

We study dynamical properties of random Schrödinger operators $H^{(ω)}$ 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 $H^{(ω)}$ to obtain the vanishing of the diffusion exponent $$ σ_{\rm diff}^+ := \limsup_{T\rightarrow\infty } \frac{\log \bbE \left(\la\la\vert X \vert^2\ra\ra_{T,f_I(H^{(ω)})ψ}\right) }{\log T}=0. $$ Here $\bbE$ is the expectation over randomness, $f_{I}$ is any smooth characteristic function of a bounded energy-interval $I$ and $ψ$ is a state vector in the domain of $H^{(ω)}$ with compact spatial support. The quantity $\la\la |X|^2 \ra\ra_{T,φ}$ denotes the Cesaro mean up to time $T$ of the second moment of position $\la |X|^2\ra_{t,φ}$ at times $0\le t\le T$ of an initial state vector $φ$. 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 Schrödinger 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.

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