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 $ \liminf_{h\to0} \|S_1(λ,h)-S_2(λ,h)\|<\sqrt{2} $ implies V1=V2, provided the fixed energy λ 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→0, 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.
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BibTeX
@misc{nicoleau2026inverse,
title = {{Inverse Semiclassical Scattering at Fixed Energy}},
author = {François Nicoleau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02175},
eprint = {2609.02175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02175},
}
RIS
TY - UNPB
TI - Inverse Semiclassical Scattering at Fixed Energy
AU - François Nicoleau
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.02175
UR - https://arxiv.org/abs/2609.02175
AB - 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 $ \liminf_{h\to0} \|S_1(λ,h)-S_2(λ,h)\|<\sqrt{2} $ implies V_1=V_2, provided the fixed energy λ 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→0, 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.
ER -