89 research outputs found
Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods
We prove a strong conditional unique continuation estimate for irreducible
quasimodes in rotationally invariant neighbourhoods on compact surfaces of
revolution. The estimate states that Laplace quasimodes which cannot be
decomposed as a sum of other quasimodes have mass bounded below by
for any on any open
rotationally invariant neighbourhood which meets the semiclassical wavefront
set of the quasimode. For an analytic manifold, we conclude the same estimate
with a lower bound of for some fixed
.Comment: 16 pages. Contains summaries of the author's results (with
co-authors) from arXiv:1103.3908, arXiv:1303.3309, and arXiv:1303.617
High-frequency resolvent estimates on asymptotically Euclidean warped products
We consider the resolvent on asymptotically Euclidean warped product
manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of
only finitely many components. We prove that the high-frequency resolvent is
either bounded by for any , or
blows up faster than any polynomial (at least along a subsequence). A stronger
result holds if the manifold is analytic. The method of proof is to exploit the
warped product structure to separate variables, obtaining a one-dimensional
semiclassical Schr\"odinger operator. We then classify the microlocal resolvent
behaviour associated to every possible type of critical value of the potential,
and translate this into the associated resolvent estimates. Weakly stable
trapping admits highly concentrated quasimodes and fast growth of the
resolvent. Conversely, using a delicate inhomogeneous blowup procedure loosely
based on the classical positive commutator argument, we show that any weakly
unstable trapping forces at least some spreading of quasimodes.
As a first application, we conclude that either there is a resonance free
region of size
for any , or there is a sequence of resonances converging to the
real axis faster than any polynomial. Again, a stronger result holds if the
manifold is analytic. As a second application, we prove a spreading result for
weak quasimodes in partially rectangular billiards.Comment: 46 pages. Contains summaries of the author's results (with
co-authors) from arXiv:1103.3908, arXiv:1303.330
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