Recovering Asymptotics of Metrics from Fixed Energy Scattering Data

The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R n from xed energy scattering data is studied. It is shown that if two such metrics, g 1 ; g 2 ; have scattering data at some xed energy which are equal up to smoothing, then there exists a diieomorphism`` xing innnity' such that g 1 g 2 is rapidly decreasing. Given the scattering matrix at two energies, it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously. These results also hold for a wide class of scattering manifolds. 1. Introduction In this paper, we examine the question of the recovery of the asymptotics of a metric from xed energy scattering data for short range perturbations of Euclidean space. We show that, modulo an inevitable diieomorphism invariance, the asymptotics are determined. This appears to be the rst result on the recovery of asymptotics of a metric, even when given the scattering matrix at all energies. Our approach is to use the techniques of 4] and invert the arising integral transforms which are considerably more complicated than those which appear there. We work in the general context of a manifold equipped with