Inverse Scattering at Fixed Energy

Let - Δ + V be a quantum mechanical two-body Hamiltonian in L2(Rn), n ≥ 3, and let S(k) be the corresponding scattering matrix at energy k2. We consider the classical problem of recovering V from knowledge of S(k) at one energy. The potential V(x) is not assumed to have any spherical symmetry. (The spherically symmetric case, including the non-uniqueness which arises if one allows potentials with reasonably mild decay at infinity, has been extensively studied—see [3] and references given there.) We show (Theorem 3.1) that if V has compact support and is in Ln/2 then it is uniquely determined by S(k); the proof gives a method to reconstruct the potential from the scattering matrix.