Microlocal properties of scattering matrices for Schr\"odinger equations on scattering manifolds

TL;DR

This paper demonstrates that scattering matrices for Schr"odinger operators on scattering manifolds are Fourier integral operators linked to geodesic flow, extending classical results to more general geometric settings.

Contribution

It establishes that scattering matrices are Fourier integral operators associated with canonical transforms generated by geodesic flow on scattering manifolds, generalizing prior Euclidean results.

Findings

Scattering matrices are Fourier integral operators.

Wave front sets are transformed by the canonical transform.

Results extend Melrose and Zworski's theorem to scattering manifolds.

Abstract

Let $M$ be a scattering manifold, i.e., a Riemannian manifold with asymptotically conic structure, and let $H$ be a Schr\"odinger operator on $M$. We can construct a natural time-dependent scattering theory for $H$ with a suitable reference system, and the scattering matrix is defined accordingly. We here show 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 2-body quantum scattering on Euclidean spaces.