Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 2 (2013), 257-286.

Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds

Kenichi Ito and Shu Nakamura

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 6, Number 2 (2013), 257-286.

Dates
Received: 20 April 2011
Revised: 28 March 2012
Accepted: 23 May 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731311

Digital Object Identifier
doi:10.2140/apde.2013.6.257

Mathematical Reviews number (MathSciNet)
MR3071392

Zentralblatt MATH identifier
1273.35201

Subjects
Primary: 35P25: Scattering theory [See also 47A40] 35S30: Fourier integral operators 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx] 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]

Keywords
scattering matrix scattering manifolds Schrödinger operators semiclassical analysis

Citation

Ito, Kenichi; Nakamura, Shu. Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds. Anal. PDE 6 (2013), no. 2, 257--286. doi:10.2140/apde.2013.6.257. https://projecteuclid.org/euclid.apde/1513731311


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