Jared Wunsch
Non-concentration of quasimodes for integrable systems.
Fri, 08/27/2010 - 15:01 — SomNambulAuthors: Jared Wunsch
Subjects: Analysis of PDEs
AbstractWe consider the possible concentration in phase space of a sequence of
eigenfunctions (or, more generally, a quasimode) of an operator whose principal
symbol has completely integrable Hamilton flow. The semiclassical wavefront set
$WF_h$ of such a sequence is invariant under the Hamilton flow.Resolvent estimates for normally hyperbolic trapped sets.
Thu, 03/25/2010 - 16:01 — SomNambulAuthors: Jared Wunsch, Maciej Zworski
Subjects: Analysis of PDEs
AbstractWe give pole free strips and estimates for resolvents of semiclassical
operators which, on the level of the classical flow, have normally hyperbolic
smooth trapped sets of codimension two in phase space. Such trapped sets are
structurally stable and our motivation comes partly from considering the wave
equation for Kerr black holes and their perturbations, whose trapped sets have
precisely this structure. We give applications including local smoothing
effects with epsilon derivative loss for the Schr\"odinger propagator as well
as local energy decay results for the wave equation.Positive commutators at the bottom of the spectrum.
Mon, 09/28/2009 - 15:24 — SomNambulAuthors: Andras Vasy, Jared Wunsch
Subjects: Analysis of PDEs
AbstractBony and H\"afner have recently obtained positive commutator estimates on the
Laplacian in the low-energy limit on asymptotically Euclidean spaces; these
estimates can be used to prove local energy decay estimates if the metric is
non-trapping. We simplify the proof of the estimates of Bony-H\"afner and
generalize them to the setting of scattering manifolds (i.e. manifolds with
large conic ends), by applying a sharp Poincar\'e inequality.
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