Recovering Asymptotics of Short Range Potentials

Abstract:

Any compact smooth manifold with boundary admits a Riemann metric of the form near the boundary, where x is the boundary defining function and h' restricts to a Riemannian metric, h, on the boundary. Melrose has associated a scattering matrix to such a metric which was shown by he and Zworski to be a Fourier integral operator. It is shown here that the principal symbol of the difference of the scattering matrices for two potentials at fixed energy determines a weighted integral of the lead term of V ₁ - V ₂ over all geodesics on the boundary. This is used to prove that the entire Taylor series of the potential at the boundary is determined by the scattering matrix at a non-zero fixed energy for certain manifolds including Euclidean space.