Localization for alloy-type models with non-monotone potentials

We consider a family of self-adjoint operators [H_ω= - Δ+ λV_ω, \quad ω\in Ω= \bigtimes_{k \in \ZZ^d} \RR,] on the Hilbert space $\ell^2 (\ZZ^d)$ or $L^2 (\RR^d)$. Here $Δ$ denotes the Laplace operator (discrete or continuous), $V_ω$ is a multiplication operator given by the function $$V_ω(x) = \sum_{k \in \ZZ^d} ω_k u(x-k) on $\ZZ^d$, or \quad V_ω(x) = \sum_{k \in \ZZ^d} ω_k U(x-k) on $\RR^d$,$$ and $λ> 0$ is a real parameter modeling the strength of the disorder present in the model. The functions $u:\ZZ^d \to \RR$ and $U:\RR^d \to \RR$ are called single-site potential. Moreover, there is a probability measure $\PP$ on $Ω$ modeling the distribution of the individual configurations $ω\in Ω$. The measure $\PP = \prod_{k \in \ZZ^d} μ$ is a product measure where $μ$ is some probability measure on $\RR$ satisfying certain regularity assumptions. The operator on $L^2 (\RR^d)$ is called alloy-type model, and its analogue on $\ell^2 (\ZZ^d)$ discrete alloy-type model. This thesis refines the methods of multiscale analysis and fractional moments in the case where the single-site potential is allowed to change its sign. In particular, we develop the fractional moment method and prove exponential localization for the discrete alloy-type model in the case where the support of $u$ is finite and $u$ has fixed sign at the boundary of its support. We also prove a Wegner estimate for the discrete alloy-type model in the case of exponentially decaying but not necessarily finitely supported single-site potentials. This Wegner estimate is applicable for a proof of localization via multiscale analysis.

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