69 research outputs found
Spectral gaps for normally hyperbolic trapping
We establish a resonance free strip for codimension 2 symplectic normally
hyperbolic trapped sets with smooth incoming/outgoing tails. An important
application is wave decay on Kerr and Kerr-de Sitter black holes. We recover
the optimal size of the strip and give an resolvent bound there. We
next show existence of deeper resonance free strips under the -normal
hyperbolicity assumption and a pinching condition. We also give a lower bound
on the ne-sided cutoff resolvent on the real line.Comment: 24 pages, 4 figures; more minor corrections. To appear in Ann. Inst.
Fourie
Microlocal limits of Eisenstein functions away from the unitarity axis
We consider a surface M with constant curvature cusp ends and its Eisenstein
functions E_j(\lambda). These are the plane waves associated to the j-th cusp
and the spectral parameter \lambda, (\Delta - 1/4 - \lambda^2)E_j = 0. We prove
that as Re\lambda \to \infty and Im\lambda \to \nu > 0, E_j converges
microlocally to a certain naturally defined measure decaying exponentially
along the geodesic flow. In particular, for a sequence of \lambda's
corresponding to scattering resonances, we find the microlocal limit of
resonant states with energies away from the real line. This statement is
similar to quantum unique ergodicity (QUE), which holds in certain other
situations; however, the proof uses only the structure of the infinite ends,
not the global properties of the geodesic flow. As an application, we also show
that the scattering matrix tends to zero in strips separated from the real
line.Comment: 21 pages; to appear in Journal of Spectral Theor
Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
This expository article, written for the proceedings of the Journ\'ees EDP
(Roscoff, June 2017), presents recent work joint with Jean Bourgain
[arXiv:1612.09040] and Long Jin [arXiv:1705.05019]. We in particular show that
eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below
in norm on each nonempty open set, by a constant depending on the set but
not on the eigenvalue.Comment: 18 page
Dynamical zeta functions for Axiom A flows
We show that the Ruelle zeta function of any smooth Axiom A flow with
orientable stable/unstable spaces has a meromorphic continuation to the entire
complex plane. The proof uses the meromorphic continuation result of
[arXiv:1410.5516] together with the work of Conley-Easton and
[arXiv:1711.10059] which imply that every basic hyperbolic set can be put into
the framework of [arXiv:1410.5516].Comment: 6+eps pages, added reference
Fractal Weyl laws and wave decay for general trapping
We prove a Weyl upper bound on the number of scattering resonances in strips
for manifolds with Euclidean infinite ends. In contrast with previous results,
we do not make any strong structural assumptions on the geodesic flow on the
trapped set (such as hyperbolicity) and instead use propagation statements up
to the Ehrenfest time. By a similar method we prove a decay statement with high
probability for linear waves with random initial data. The latter statement is
related heuristically to the Weyl upper bound. For geodesic flows with positive
escape rate, we obtain a power improvement over the trivial Weyl bound and
exponential decay up to twice the Ehrenfest time.Comment: 36 pages, 5 figures; minor revision
Semiclassical measures on hyperbolic surfaces have full support
We show that each limiting semiclassical measure obtained from a sequence of
eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on
the entire cosphere bundle. The key new ingredient for the proof is the fractal
uncertainty principle, first formulated in [arXiv:1504.06589] and proved for
porous sets in [arXiv:1612.09040].Comment: 39 pages, 3 figures; changed following the referees' suggestions. To
appear in Acta Mat
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