The resolvent for Laplace-type operators on asymptotically conic spaces

Research Report MRR00-008

The resolvent for Laplace-type operators on asymptotically conic spaces

Andrew Hassell and András Vasy

Abstract: Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which may be either of short range or `gravitational' long range type. Thus, $g$ gives $X$ the geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be an operator of the form $H = \Delta + P$, where $\Delta$ is the Laplacian with respect to $g$ and $P$ is a self-adjoint first order scattering differential operator with coefficients vanishing at $\partial X$ and satisfying a `gravitational' condition. We define a symbol calculus for Legendre distributions on manifolds with codimension two corners and use it to give a direct construction of the resolvent kernel of $H$, $R(\sigma + i0)$, for $\sigma$ on the positive real axis. In this approach, we do not use the limiting absorption principle at any stage; instead we construct a parametrix which solves the resolvent equation up to a compact error term and then use Fredholm theory to remove the error term.

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