Inverse Semiclassical Scattering at Fixed Energy

Abstract

We investigate inverse scattering at a fixed energy for semiclassical Schrödinger operators with smooth potentials. For compactly supported potentials, we show that, at a non-trapping energy, if the corresponding semiclassical scattering matrices differ by $o(1)$ in operator norm as $h\to0$, then the associated classical scattering maps coincide. The proof relies on Ingremeau's description of the action of the scattering matrix on coherent states. Combined with a classical rigidity result of Stefanov--Uhlmann--Vasy, this yields uniqueness of the potential under a natural virial condition, provided the energy lies above the potential. We then consider radial short-range repulsive potentials. Under a monotonicity assumption on the radial force, the classical scattering relation has a single branch. Combining the semiclassical scattering asymptotics of Robert--Tamura with the classical Firsov--Abel inversion formula, we show that the differential cross section at a single fixed energy determines the potential throughout the classically accessible region. Finally, we obtain an analogous rigidity result for simple compactly supported metric perturbations.