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 satisfying a natural virial condition, we prove that, under the sharp condition, ${\mathrm{liminf}}_{h\to 0}\Vert S_1(\lambda ,h)-S_2(\lambda ,h)\Vert <2$ 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 fixed energy, up to $o(1)$ as $h\to 0$, determines the potential in 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.

arXiv commentThe upper bound constant \(\sqrt{2}\) has been replaced by the optimal constant \(2\)