Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on #(H 0 ) with probability one. To prove this result we combine Aizenman-Molchanov theory with techniques of scattering theory.

We present a new, short, self-contained proof of localization properties of multi-dimensional continuum random Schrödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2], but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.

We consider continuum random Schrödinger operators of the type Hω = −∆+V0 + Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ +V0. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (“random tube”) in arbitrary dimension.

We discuss spectral and scattering theory of the discrete Laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics. 1.

Abstract. We prove Anderson localization and strong dynamical localization for random surface models in R d. 1. Introduction, the model, and the results. Spectral and scattering theory for mathematical models of rough surfaces has attracted considerable interest in recent years, as witnessed in [2, 3, 7–13, 16]. One of the reasons is that these models exhibit a metal-insulator transition. This transition is expected for typical random models in dimensions three and higher but, unfortunately, a