Unbounded potential recovery  in the plane

Astala, Kari; Faraco, Daniel; Rogers, Keith M.

doi:10.24033/asens.2302

Unbounded potential recovery in the plane
[Reconstruction de potentiels non bornés dans le plan]

Astala, Kari ; Faraco, Daniel ; Rogers, Keith M.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1027-1051.

Résumé
Abstract

Nous reconstruisons des potentiels à support compact avec une demi-derivée dans $L^{2}$ à partir de l'amplitude de diffusion à énergie fixe. Pour cela, nous établissons un lien entre une méthode récemment introduite par Bukhgeim pour déterminer de façon unique le potentiel à partir de l'application Dirichlet-to-Neumann, et une question de Carleson qui concerne la convergence vers la donnée initiale des solutions de l'équation de Schrödinger dépendante du temps. Nous fournissons également des exemples de potentiels à support compact, avec $s$ dérivées dans $L^{2}$ pour tout $s < 1 / 2$ , qui ne peuvent pas être reconstruits par cette méthode. Ainsi, la méthode de reconstruction a un seuil en termes de la régularité qui diffère du résultat d'unicité.

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.

Publié le : 2016-11-12

DOI : 10.24033/asens.2302

Classification : 35P25, 45Q05; 42B37, 35J10.
Keywords: Inverse problems, scattering, almost everywhere convergence, Schrödinger.
Mot clés : Problèmes inverses, théorie de la diffusion, convergence presque partout, Schrödinger.

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Astala, Kari; Faraco, Daniel; Rogers, Keith M. Unbounded potential recovery  in the plane. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1027-1051. doi : 10.24033/asens.2302. http://www.numdam.org/articles/10.24033/asens.2302/

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