Rough Potential Recovery in the Plane

TL;DR

This paper investigates the problem of reconstructing compactly supported potentials in the plane from fixed-energy scattering data, establishing regularity thresholds for successful recovery and connecting it to the Bukhgeim method and Carleson’s question.

Contribution

It introduces a new regularity threshold for potential recovery using the Bukhgeim method and relates it to Carleson’s convergence question for Schrödinger equations.

Findings

Potential can be reconstructed with half a derivative in L^2.

Potentials with less than half a derivative in L^2 cannot be recovered.

The recovery method has a different regularity threshold than the uniqueness result.

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.