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 Rⁿ from fixed energy scattering data is studied. It is shown that if two such metrics, g₁,g₂, have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism N 'fixing infinity' such that N^*g₁-g₂ 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.