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Resolvent at low energy III: The spectral measure

Authors: Colin Guillarmou, Andrew Hassell and Adam Sikora
Journal: Trans. Amer. Math. Soc. 365 (2013), 6103-6148
MSC (2010): Primary 35P25, 47A40, 58J50
Published electronically: April 2, 2013
MathSciNet review: 3091277
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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) \big ( R(\lambda +i0) - R(\lambda - i0) \big )$ of $ P_+^{1/2}$, where $ R(\lambda ) = (P - \lambda ^2)^{-1}$, as $ \lambda \to 0$, in a manner similar to that done by the second author and Vasy (2001) and by the first two authors (2008, 2009). The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which was introduced in the 2008 work of the first two authors and is obtained from $ M^2 \times [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 Schrödinger propagator $ e^{itP_+}$, as $ t \to \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 Schrödinger equation on $ M$, provided $ M$ is nontrapping.

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Additional Information

Colin Guillarmou
Affiliation: DMA, U.M.R. 8553 CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, F 75230 Paris cedex 05, France

Andrew Hassell
Affiliation: Department of Mathematics, Australian National University, Canberra ACT 0200, Australia

Adam Sikora
Affiliation: Department of Mathematics, Australian National University, Canberra ACT 0200, Australia — and — Department of Mathematics, Macquarie University, NSW 2109, Australia

Keywords: Scattering metric, asymptotically conic manifold, resolvent kernel, spectral measure, low energy asymptotics, Price's law.
Received by editor(s): September 16, 2010
Received by editor(s) in revised form: April 4, 2012
Published electronically: April 2, 2013
Additional Notes: The second and third authors were supported by Australian Research Council Discovery grants DP0771826 and DP1095448 and the second author by a Future Fellowship. The first author was partially supported by ANR grant ANR-09-JCJC-0099-01 and by the PICS-CNRS Progress in Geometric Analysis and Applications, and thanks the math department of ANU for its hospitality. The first author also thanks M.Tohaneanu for useful discussions.
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