Inverse Semiclassical Scattering at Fixed Energy

Abstract

We investigate inverse scattering at a fixed energy for semiclassical Schr\"odinger operators with smooth potentials. For compactly supported potentials satisfying a natural virial condition, we prove that $ \liminf_{h\to0} \|S_1(\lambda,h)-S_2(\lambda,h)\|<\sqrt{2} $ implies $V_1=V_2$, provided the fixed energy $\lambda$ lies above both potentials. We then consider radial short-range repulsive potentials. Under a monotonicity assumption on the radial force, we show that the semiclassical differential cross section at a single fixed energy, up to $o(1)$ as $h\to0$, determines the potential throughout the classically accessible region. Finally, we obtain an analogous rigidity result for nontrapping compactly supported perturbations of the Euclidean metric under a strict convexity assumption.