Vol. 6, No. 2, 2013

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Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds

Kenichi Ito and Shu Nakamura

Vol. 6 (2013), No. 2, 257–286
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.

Keywords
scattering matrix, scattering manifolds, Schrödinger operators, semiclassical analysis
Mathematical Subject Classification 2010
Primary: 35P25, 35S30, 58J40, 58J50
Milestones
Received: 20 April 2011
Revised: 28 March 2012
Accepted: 23 May 2012
Published: 24 June 2013
Authors
Kenichi Ito
Graduate School of Pure and Applied Sciences
University of Tsukuba
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
Japan
Shu Nakamura
Graduate School of Mathematical Sciences
University of Tokyo
3-8-1, Komaba
Meguro, Tokyo 153-8914
Japan
http://www.ms.u-tokyo.ac.jp/~shu/