Discrete Schrödinger operators with random alloy-type potential

Abstract

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schrödinger operators $H_{\omega }=-\Delta +V_{\omega }$ on $\ell^2 (\ZZ^d)$ where $\Delta$ is the discrete Laplacian and $V_{\omega }$ the multiplication by the function $V_ω(x) = \sum_{k \in \ZZ^d} ω_k u(x-k)$. Here ${\omega }_k$, $k \in \ZZ^d$, are i.i.d. random variables and $u \in \ell^1 (\ZZ^d ; \RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_{\omega }$ depend in a non-monotone way on the random parameters ${\omega }_k$. This requires new methods at certain stages of the localization proof.

Proceedings of the Spectral Days 2010, Pontificia Universidad Católica de Chile, Santiago