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Microlocal properties
of scattering matrices for Schrödinger equations on
scattering manifolds
Kenichi Ito and Shu Nakamura |
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Vol. 6 (2013), No. 2, 257–286
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Abstract |
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Let be a scattering manifold, i.e., a Riemannian manifold with an asymptotically conic structure, and let be a Schrödinger operator on . One can construct a natural time-dependent scattering theory for 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
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Mathematical Subject Classification 2010
Primary: 35P25, 35S30, 58J40, 58J50
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Milestones
Received: 20 April 2011
Revised: 28 March 2012
Accepted: 23 May 2012
Published: 24 June 2013
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Authors |
| Kenichi Ito | |
| Graduate School of Pure and Applied
Sciences University of Tsukuba 1-1-1 Tennodai Tsukuba, Ibaraki 305-8571 Japan |
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| Shu Nakamura | |
| Graduate School of Mathematical
Sciences University of Tokyo 3-8-1, Komaba Meguro, Tokyo 153-8914 Japan |
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| http://www.ms.u-tokyo.ac.jp/~shu/ | |
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