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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2013

Resolvent at low energy III: the spectral measure

Résumé

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 in \cite{HV2}, and by the first two authors in \cite{GH1, GH2}. The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which was introduced in \cite{GH1} 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.
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Dates et versions

hal-00518002 , version 1 (16-09-2010)

Identifiants

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Colin Guillarmou, Andrew Hassell, Adam Sikora. Resolvent at low energy III: the spectral measure. Transactions of the American Mathematical Society, 2013, 365, pp.6103-6148. ⟨10.1090/S0002-9947-2013-05849-7⟩. ⟨hal-00518002⟩
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