Scattering for Schroedinger operators with potentials concentrated near a subspace
In this talk, I will consider the properties ofSchroedinger operators with potentials concentrated near a subspace of R^d.This is one of many models of a quantum particle interacting with a surface.For such a system, one would expect that states that move away from thepotential should behave asymptotically like a free state. So a natural questionis, are all states asymptotically free? And if not, can wecharacterize the states that are not asymptotically free? Our main resultsutilize a novel interpretation of the Enss method to obtain a dynamicalcharacterization of the orthogonal complement of the asymptotically freestates. This orthogonal complement is a set of ``surface states," whichconsists of states confined to the subspace (such as pure point states) andstates that escape it at a sublinear rate, in a suitable sense.
In this talk, I will state our results, sketch some of themain ideas in the proof, and briefly discuss examples of these surface statesfor different systems. This is joint work with Adam Black.