Resolvent at low energy III: the spectral measure

Understand

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 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 $M^2 \times [0, \lambda_0)$ by blowing up a certain number of boundary faces.