We consider a discrete Schrodinger operator H = +V acting in l 2 (Z d ), with periodic potential V supported by the subspace "surface" f0g Z d2 . We prove that the spectrum of H is purely absolutely continuous, and that surface waves (see [8] for denition) oscillate in the longitudinal directions to the "surface". We nd also an explicit formula for the generalized spectral shift function introduced in [4]. 1 Introduction In this paper we will primarily discuss the discrete Schrodinger operator H with a surface potential V acting on the Hilbert space l 2 (Z d ) H = H 0 + V; (1.1) V (X) = (x)v(); (1.2) where Z d = Z d 1 Z d 2 = fX = (x; ) j x 2 Z d 1 ; 2 Z d 2 g; (1.3) In other words H (X) = X Y 2Z d ;jY Xj=1 (Y ) + (x)v() (X); (1.4) for all 2 l 2 (Z d ), where (x) is Kronecker symbol. 1 It is well known that H 0 is a bounded self-adjoint operator on l 2 (Z d ), and ac (H 0 ) = (H 0 ) = [ 2d; 2d]; pp (H 0 ) = sc (H 0 ) = ...

We prove a Wegner estimate for generalized alloy type models at negative energies (Theorems 8 and 13). The single site potential is assumed to be non- positive. The random potential does not need to be stationary with respect to translations from a lattice. Actually, the set of points to which the individual single site potentials are attached, needs only to satisfy a certain density condition. The distribution of the coupling constants is assumed to have a bounded density only in the energy region where we prove the Wegner estimate.

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.

ABSTRACT. We consider Schrödinger operators on L 2 (R d) with a random potential concentrated near the surface R d1 × {0} ⊂ R d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail rlies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of

Abstract. This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered: fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably “reduced ” to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.

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.

Let G be a simple countable connected graph and let H 0 be the discrete Laplacian on l (G). Let G and let V = P n2 V (n)( n j ) n be a potential supported on .

(Communicated by Mikhail Shubin) Abstract. It is well known that the Green function of the standard discrete Laplacian on l2 (Zd), ∆st ψ(n) =(2d) −1 ψ(m),

Abstract. We prove that the spectrum of a Schrödinger operator with a potential which is periodic in certain directions and superexponentially decaying in the others is purely absolutely continuous. Therefore, we reduce the operator using the Bloch-Floquet-Gelfand transform in the periodic variables, and show that, except for at most a set of quasi-momenta of measure zero, the reduced operators satisfies a limiting absorption principle.