93 research outputs found
Semiclassical resolvent estimates at trapped sets
We extend our recent results on propagation of semiclassical resolvent
estimates through trapped sets when a priori polynomial resolvent bounds hold.
Previously we obtained non-trapping estimates in trapping situations when the
resolvent was sandwiched between cutoffs microlocally supported away from the
trapping, a microlocal version of a result of Burq and Cardoso-Vodev. We now
allow one of the two cutoffs to be supported at the trapped set, giving
estimates which are intermediate between the trapping and non-trapping ones.Comment: 5 page
Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes
We consider the wave equation on asymptotically Minkowski spacetimes and the
Klein-Gordon equation on even asymptotically de Sitter spaces. In both cases we
show that the extreme difference of propagators (i.e. retarded propagator minus
advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces
a symplectic form on the space of solutions with wave front set confined to the
radial sets. Furthermore, we construct isomorphisms between the solution spaces
and symplectic spaces of asymptotic data. As an application of this result we
obtain distinguished Hadamard two-point functions from asymptotic data.
Ultimately, we prove that the corresponding Quantum Field Theory on
asymptotically de Sitter spacetimes induces canonically a QFT beyond the future
and past conformal boundary, i.e. on two even asymptotically hyperbolic spaces.
Specifically, we show this to be true both at the level of symplectic spaces of
solutions and at the level of Hadamard two-point functions.Comment: 53 p., 4 figures; introduction and App. A.2 expanded, minor
improvements; to appear in Ann. Henri Poincar\'
Propagation of singularities around a Lagrangian submanifold of radial points
In this work we study the wavefront set of a solution u to Pu = f, where P is
a pseudodifferential operator on a manifold with real-valued homogeneous
principal symbol p, when the Hamilton vector field corresponding to p is radial
on a Lagrangian submanifold contained in the characteristic set of P. The
standard propagation of singularities theorem of Duistermaat-Hormander gives no
information at the Lagrangian submanifold. By adapting the standard
positive-commutator estimate proof of this theorem, we are able to conclude
additional regularity at a point q in this radial set, assuming some regularity
around this point. That is, the a priori assumption is either a weaker
regularity assumption at q, or a regularity assumption near but not at q.
Earlier results of Melrose and Vasy give a more global version of such
analysis. Given some regularity assumptions around the Lagrangian submanifold,
they obtain some regularity at the Lagrangian submanifold. This paper
microlocalizes these results, assuming and concluding regularity only at a
particular point of interest. We then proceed to prove an analogous result,
useful in scattering theory, followed by analogous results in the context of
Lagrangian regularity.Comment: 39 pages, 4 figure
Propagation of singularies in three-body scattering
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.Includes bibliographical references (p. 106-107).by András Vasy.Ph.D
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