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Rough Potential Recovery in the Plane
- Source :
- Ann. Sci. \'Ec. Norm. Sup\'er. (4) 49 (2016), no. 5, 1027-1051
- Publication Year :
- 2013
-
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.<br />Comment: 24 pages
Details
- Database :
- arXiv
- Journal :
- Ann. Sci. \'Ec. Norm. Sup\'er. (4) 49 (2016), no. 5, 1027-1051
- Publication Type :
- Report
- Accession number :
- edsarx.1304.1317
- Document Type :
- Working Paper