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 give a number of results concerning dierent possible spectral types for Schrodinger operators with sparse potentials. These potentials are in between stationary (e.g., random) potentials and the short range potentials familiar from scattering theory. They decay at in nity in some averaged sense, however in such a way that there is enough \space" for surprising spectral properties.