11,972 research outputs found
Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends
We prove some sharp upper bounds on the number of resonances associated with
the Laplacian, or Laplacian plus potential, on a manifold with infinite
cylidrical ends
Asymptotics for a resonance-counting function for potential scattering on cylinders
We study certain resonance-counting functions for potential scattering on
infinite cylinders or half-cylinders. Under certain conditions on the
potential, we obtain asymptotics of the counting functions, with an explicit
formula for the constant appearing in the leading term
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
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