Rough Potential Recovery in the Plane

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 Schrödinger 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.

24 pages

References in corpus (4)

• Uniqueness in Calderon's problem with Lipschitz conductivities
• Calderon inverse Problem with partial data on Riemann Surfaces
• Identification of a connection from Cauchy data on a Riemann surface with boundary
• On the Schrodinger maximal function in higher dimension