Resolvent at low energy III: The spectral measure

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.