10,352 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
Resonances for Schr\"odinger operators on infinite cylinders and other products
We study the resonances of Schr\"odinger operators on the infinite product
, where is odd, is the
unit circle, and the potential . This paper shows that at
high energy, resonances of the Schr\"odinger operator on
which are near the continuous spectrum are
approximated by the resonances of on , where the potential
given by averaging over the unit circle. These resonances are, in
turn, given in terms of the resonances of a Schr\"odinger operator on
which lie in a bounded set. If the potential is smooth, we
obtain improved localization of the resonances, particularly in the case of
simple, rank one poles of the corresponding scattering resolvent on
. In that case, we obtain the leading order correction for the
location of the corresponding high energy resonances. In addition to direct
results about the location of resonances, we show that at high energies away
from the resonances, the resolvent of the model operator on
approximates that of on . If , in certain cases this
implies the existence of an asymptotic expansion of solutions of the wave
equation. Again for the special case of , we obtain a resonant rigidity
type result for the zero potential among all real-valued potentials.Comment: 46 pages; v. 2 is attempt to fix uploading erro
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