2007年11月21日
Singularities of solutions to Schrodinger equation on scattering manifold
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In this paper we study microlocal singularities of solutions to Schrodinger<br />
equations on scattering manifolds, i.e., noncompact Riemannian manifolds with<br />
asymptotically conic ends. We characterize the wave front set of the solutions<br />
in terms of the initial condition and the classical scattering maps under the<br />
nontrapping condition. Our result is closely related to a recent work by<br />
Hassell and Wunsch, though our model is more general and the method, which<br />
relies heavily on scattering theoretical ideas, is simple and quite different.<br />
In particular, we use Egorov-type argument in the standard pseudodifferential<br />
symbol classes, and avoid using Legendre distributions. In the proof, we employ<br />
a microlocal smoothing property in terms of the radially homogenous wave front<br />
set, which is more precise than the preceding results.
equations on scattering manifolds, i.e., noncompact Riemannian manifolds with<br />
asymptotically conic ends. We characterize the wave front set of the solutions<br />
in terms of the initial condition and the classical scattering maps under the<br />
nontrapping condition. Our result is closely related to a recent work by<br />
Hassell and Wunsch, though our model is more general and the method, which<br />
relies heavily on scattering theoretical ideas, is simple and quite different.<br />
In particular, we use Egorov-type argument in the standard pseudodifferential<br />
symbol classes, and avoid using Legendre distributions. In the proof, we employ<br />
a microlocal smoothing property in terms of the radially homogenous wave front<br />
set, which is more precise than the preceding results.
- ID情報
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- arXiv ID : arXiv:0711.3258