Scattering for Schr\"{o}dinger operators with potentials concentrated near a subspace

TL;DR

This paper investigates the scattering behavior of Schr"odinger operators with potentials localized near a subspace, identifying surface states and their properties using a novel Enss method interpretation.

Contribution

It introduces a new approach to characterize surface states and their dynamics for Schr"odinger operators with subspace-concentrated potentials.

Findings

Existence of scattering states for these operators.

Characterization of surface states as confined or slowly escaping.

Examples demonstrating different types of surface states.

Abstract

We study the scattering properties of Schr\"{o}dinger operators with bounded potentials concentrated near a subspace of $\mathbb{R}^d$. 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 in order to obtain a dynamical characterisation of the orthogonal complement of the scattering states.