53 research outputs found
Propagation of singularities for the wave equation on manifolds with corners
In this paper we describe the propagation of smooth (C^\infty) and Sobolev
singularities for the wave equation on smooth manifolds with corners M equipped
with a Riemannian metric g. That is, for X=MxR, P=D_t^2-\Delta_M, and u locally
in H^1 solving Pu=0 with homogeneous Dirichlet or Neumann boundary conditions,
we show that the wave front set of u is a union of maximally extended
generalized broken bicharacteristics. This result is a smooth counterpart of
Lebeau's results for the propagation of analytic singularities on real analytic
manifolds with appropriately stratified boundary. Our methods rely on
b-microlocal positive commutator estimates, thus providing a new proof for the
propagation of singularities at hyperbolic points even if M has a smooth
boundary (and no corners)
Semiclassical estimates in asymptotically Euclidean scattering
In this note we obtain semiclassical resolvent estimates for non-trapping
long range perturbations of the Laplacian on asymptotically Euclidean
manifolds. Our proof is based on a positive commutator argument which differs
from Mourre-type estimates by making the commutant also positive. The resolvent
estimates, including the weighting of the Sobolev spaces in the estimates, are
an immediate consequence.Comment: 11 pages, 4 figures, AMS Late
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