Weyl Asymptotics for the Laplacian on Asymptotically Euclidean Spaces

Abstract. We prove a trace-type formula for the Laplacian on an asymptotically Euclidean space and use this to obtain Weyl asymptotics of the scattering phase. Other results include an explicit calculation of the leading order singularity of the scattering matrix and results on the behavior of the resolvent and scattering matrix near 0. In this paper we obtain a relative trace formula for the Laplacian on an asymp-totically Euclidean space. The existence of such a formula involving the scattering phase was conjectured in [22]. We use this formula, some techniques of Robert, and some microlocal analysis to obtain Weyl asymptotics for the scattering phase. We also compute explicitly the leading singularity of the (absolute) scattering ma-trix. Let X be a smooth, compact, n-dimensional manifold with boundary @X, equipped with a scattering metric g. That is, g is a smooth metric on the interior of X which, near the boundary, can be brought to the form g = dx2 x4 h x