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Singularities of solutions to the Schrödinger equation on scattering manifold
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 6, December 2009
- pp. 1835-1865
- 10.1353/ajm.0.0087
- Article
- Additional Information
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In this paper we study microlocal singularities of solutions to Schr\"odinger equations on
scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We
characterize the wave front set of the solutions in terms of the initial condition and the
classical scattering maps under the nontrapping condition. Our result is closely related to a
recent work by Hassell and Wunsch, though our model is more general and the method, which relies
heavily on scattering theoretical ideas, is simple and quite different. In particular, we use an
Egorov-type argument in the standard pseudodifferential symbol classes, and avoid using Legendre
distributions. In the proof, we employ a microlocal smoothing property in terms of the {\it
radially homogenous wave front set}, which is more precise than the preceding results.