Unbounded potential recovery in the plane

Kari Astala, Daniel Faraco, Keith M. Rogers

Research output: Contribution to journalArticleScientificpeer-review

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 Schrodinger 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.
Original languageEnglish
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume49
Issue number5
Pages (from-to)1027-1051
Number of pages25
ISSN0012-9593
Publication statusPublished - Sep 2016
MoE publication typeA1 Journal article-refereed

Fields of Science

  • 111 Mathematics

Cite this

Astala, Kari ; Faraco, Daniel ; Rogers, Keith M. / Unbounded potential recovery in the plane. In: Annales Scientifiques de l'Ecole Normale Superieure. 2016 ; Vol. 49, No. 5. pp. 1027-1051.
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Unbounded potential recovery in the plane. / Astala, Kari; Faraco, Daniel; Rogers, Keith M.

In: Annales Scientifiques de l'Ecole Normale Superieure, Vol. 49, No. 5, 09.2016, p. 1027-1051.

Research output: Contribution to journalArticleScientificpeer-review

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AB - 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 Schrodinger 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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