Surface states and spectra

Let ℤ^d+1₊ = ℤ^d × ℤ₊, let H₀ be the discrete Laplacian on the Hilbert space l²(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 = H₀+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 σ(H₀) with probability one. To prove this result we combine Aizenman-Molchanov theory with techniques of scattering theory.