Resolvent at low energy III: The spectral measure

Abstract

Let M◦ be a complete noncompact manifold and g an asymptotically conic Riemaniann metric on M◦, in the sense that M◦ compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. Let Δ be the positive Laplacian associated to g, and P = Δ+ V , where V is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure dE(λ)=(λ/πi)

R(λ + i0) − R(λ − i0) of P+¹/² , where R(λ)=(P − λ²)⁻¹, as λ → 0, in a manner similar to that done by the second author and Vasy (2001) and by the first two authors (2008, 2009). Show more