32 research outputs found
Wave asymptotics for waveguides and manifolds with infinite cylindrical ends
We describe wave decay rates associated to embedded resonances and spectral
thresholds for waveguides and manifolds with infinite cylindrical ends. We show
that if the cut-off resolvent is polynomially bounded at high energies, as is
the case in certain favorable geometries, then there is an associated
asymptotic expansion, up to a remainder, of solutions of the wave
equation on compact sets as . In the most general such case we
have , and under an additional assumption on the infinite ends we have
. If we localize the solutions to the wave equation in frequency
as well as in space, then our results hold for quite general waveguides and
manifolds with infinite cylindrical ends.
To treat problems with and without boundary in a unified way, we introduce a
black box framework analogous to the Euclidean one of Sj\"ostrand and Zworski.
We study the resolvent, generalized eigenfunctions, spectral measure, and
spectral thresholds in this framework, providing a new approach to some mostly
well-known results in the scattering theory of manifolds with cylindrical ends.Comment: In this revision we work in a more general black box setting than in
the first version of the paper. In particular, we allow a boundary extending
to infinity. The changes to the proofs of the main theorems are minor, but
the presentation of the needed basic material from scattering theory is
substantially expanded. New examples are included, both for the main results
and for the black box settin
Low energy scattering asymptotics for planar obstacles
We compute low energy asymptotics for the resolvent of a planar obstacle, and
deduce asymptotics for the corresponding scattering matrix, scattering phase,
and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to
relate the obstacle resolvent to the free resolvent and an identity of Petkov
and Zworski to relate the scattering matrix to the resolvent. The leading
singularities are given in terms of the obstacle's logarithmic capacity or
Robin constant. We expect these results to hold for more general compactly
supported perturbations of the Laplacian on , with the definition
of the Robin constant suitably modified, under a generic assumption that the
spectrum is regular at zero.Comment: 26 pages, 1 figur
Resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides
Using coordinates , we introduce
the notion that an unbounded domain in is star shaped with
respect to . For such domains, we prove estimates on the
resolvent of the Dirichlet Laplacian near the continuous spectrum. When the
domain has infinite cylindrical ends, this has consequences for wave decay and
resonance-free regions. Our results also cover examples beyond the star-shaped
case, including scattering by a strictly convex obstacle inside a straight
planar waveguide.Comment: 21 pages, 5 figure
Semiclassical resolvent bounds for compactly supported radial potentials
We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V(|x|)−E in dimension n≥2, where h,E>0, and V:[0,∞)→ℝ is L∞ and compactly supported. We show that the weighted resolvent estimate grows no faster than exp(Ch−1), and prove an exterior weighted estimate which grows ∼h−1
Shape from Sound: Toward New Tools for Quantum Gravity
To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at least in the case of the Euclidean signature, is the discipline of spectral geometry. It aims at describing curved manifolds in terms of the spectra of their canonical differential operators. As an immediate benefit, this would offer a clean gauge-independent identification of the metric’s degrees of freedom in terms of invariants that should be ready to quantize. However, spectral geometry is itself hard and has been plagued by ambiguities. Here, we regularize and break up spectral geometry into small, finite-dimensional and therefore manageable steps. We constructively demonstrate that this strategy works at least in two dimensions. We can now calculate the shapes of two-dimensional objects from their vibrational spectra
Resolvent estimates for normally hyperbolic trapped sets
We 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.Comment: Further changes to erratum correcting small problems with Section 3.5
and Lemma 4.1; this now also corrects hypotheses, explicitly requiring
trapped set to be symplectic. Erratum follows references in this versio
Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping
We consider an -dimensional spherically symmetric, asymptotically
Euclidean manifold with two ends and a codimension 1 trapped set which is
degenerately hyperbolic. By separating variables and constructing a
semiclassical parametrix for a time scale polynomially beyond Ehrenfest time,
we show that solutions to the linear Schr\"odiner equation with initial
conditions localized on a spherical harmonic satisfy Strichartz estimates with
a loss depending only on the dimension and independent of the degeneracy.
The Strichartz estimates are sharp up to an arbitrary loss. This is
in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a
sharp local smoothing estimate with loss depending only on the degeneracy of
the trapped set, independent of the dimension
Spectral problems in open quantum chaos
This review article will present some recent results and methods in the study
of 1-particle quantum or wave scattering systems, in the semiclassical/high
frequency limit, in cases where the corresponding classical/ray dynamics is
chaotic. We will focus on the distribution of quantum resonances, and the
structure of the corresponding metastable states. Our study includes the toy
model of open quantum maps, as well as the recent quantum monodromy operator
method.Comment: Compared with the previous version, misprints and typos have been
corrected, and the bibliography update