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arXiv · 1304.1317

Rough Potential Recovery in the Plane

Kari Astala·Daniel Faraco·Keith M. Rogers
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Abstract

We reconstruct compactly supported potentials with only half a derivative in $L^2$ from the scattering amplitude at a fixed energy. For this we draw a connection between the recently introduced method of Bukhgeim, which uniquely determined the potential from the Dirichlet-to-Neumann map, and a question of Carleson regarding the convergence to initial data of solutions to time-dependent Schr\"odinger equations. We also provide examples of compactly supported potentials, with $s$ derivatives in $L^2$ for any $s<1/2$, which cannot be recovered by these means. Thus the recovery method has a different threshold in terms of regularity than the corresponding uniqueness result.

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Kari Astala, Daniel Faraco, Keith M. Rogers. 2013-04-04. Rough Potential Recovery in the Plane. https://arxiv.org/abs/1304.1317

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