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 definin g function and h' restricting to a metric on the boundary. Melrose [Spectra l and scattering theory for the Laplacian on asymptotically Euclidean space s, in Spectral and Scattering Theory, M. Ikawa, ed., Marcel Dekker, New Yor k, 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)) th at this scattering matrix is a Fourier integral operator and that the asymp totics of such potentials can be recovered from the scattering matrix for v arious manifolds including Euclidean space.