Dynamical properties of random Schr\"odinger operators

TL;DR

This paper investigates the dynamical behavior of random Schr"odinger operators, establishing conditions for localization and absence of diffusion, with applications to various models including Anderson-type and correlated potentials.

Contribution

It provides new sufficient conditions for vanishing diffusion and dynamical localization in random Schr"odinger operators, extending previous multi-scale analysis results.

Findings

Vanishing of the diffusion exponent under certain conditions

Dynamical localization can be proved for lattice models

Absence of diffusion established for weaker assumptions

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.