Scattering for Schrödinger operators with potentials concentrated near a subspace

Abstract

We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of R d \mathbb {R}^d . For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method [Comm. Math. Phys. 61 (1978), pp. 285–291] in order to obtain a dynamical characterization of the orthogonal complement of the scattering states.