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 $\hat{I}”$ be the positive Laplacian associated to $g$, and $P=\hat{I}”+V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(\hat{I}»)=(\hat{I}»/\ddot{I}€i)(R(\hat{I}»+i0)-R(\hat{I}»-i0))$ of $P_{+}^{1/2}$, where $R(\hat{I}»)=(P-\hat{I}{»}^2{)}^{-1}$, as $\hat{I}»\to 0$, 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 $M^2\times [0,\hat{I}{»}_0)$ 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. This result on the spectral measure has been used in a follow-up work by the authors (arXiv:1012.3780) to prove sharp restriction and spectral multiplier theorems on asymptotically conic manifolds.

42 pages, 4 figures