Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds

Abstract

Let $M$ be a scattering manifold, i.e., a Riemannian manifold with an asymptotically conic structure, and let $H$ be a Schrödinger operator on $M$. One can construct a natural time-dependent scattering theory for $H$ with a suitable reference system, and a scattering matrix is defined accordingly. We show here that the scattering matrices are Fourier integral operators associated to a canonical transform on the boundary manifold generated by the geodesic flow. In particular, we learn that the wave front sets are mapped according to the canonical transform. These results are generalizations of a theorem by Melrose and Zworski, but the framework and the proof are quite different. These results may be considered as generalizations or refinements of the classical off-diagonal smoothness of the scattering matrix for two-body quantum scattering on Euclidean spaces.