Recovering asymptotics of Coulomb-like potentials from fixed energy scattering data

Any compact smooth manifold, X, with boundary admits a Riemannian metric of the form x(?4)dx(2) + x(-2) h' near the boundary with x a boundary defining function and h' restricting to a metric on the boundary. Melrose [Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, in Spectral and Scattering Theory, M. Ikawa, ed., Marcel Dekker, New York, 1994] has associated a scattering matrix to such metrics and potentials in xC(infinity) (X). It is shown for potentials of the form Ax + O(x(2)) that this scattering matrix is a Fourier integral operator and that the asymptotics of such potentials can be recovered from the scattering matrix for various manifolds including Euclidean space.