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Article overview
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Resolvent at low energy III: the spectral measure | | Colin Guillarmou
; Andrew Hassell
; Adam Sikora
; | | Date: |
16 Sep 2010 | | Abstract: | Let $M^circ$ be a complete noncompact manifold and $g$ an asymptotically
conic Riemaniann metric on $M^circ$, in the sense that $M^circ$ compactifies
to a manifold with boundary $M$ in such a way that $g$ becomes a scattering
metric on $M$. Let $Delta$ be the positive Laplacian associated to $g$, and $P
= Delta + V$, where $V$ is a potential function obeying certain conditions. We
analyze the asymptotics of the spectral measure $dE(lambda) = (lambda/pi i)
ig(R(lambda+i0) - R(lambda - i0) ig)$ of $P_+^{1/2}$, where $R(lambda) =
(P - lambda^2)^{-1}$, as $lambda o 0$, in a manner similar to that done
previously by the second author and Vasy, and by the first two authors. The
main result is that the spectral measure has a simple, ’conormal-Legendrian’
singularity structure on a space which is obtained from $M^2 imes [0,
lambda_0)$ by blowing up a certain number of boundary faces. We use this to
deduce results about the asymptotics of the wave solution operators $cos(t
sqrt{P_+})$ and $sin(t sqrt{P_+})/sqrt{P_+}$, and the Schr"odinger
propagator $e^{itP}$, as $t o infty$. In particular, we prove the analogue
of Price’s law for odd-dimensional asymptotically conic manifolds.
In future articles, this result on the spectral measure will be used to (i)
prove restriction and spectral multiplier estimates on asymptotically conic
manifolds, and (ii) prove long-time dispersion and Strichartz estimates for
solutions of the Schr"odinger equation on $M$, provided $M$ is nontrapping. | | Source: | arXiv, 1009.3084 | | Services: | Forum | Review | PDF | Favorites | |
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