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Annales scientifiques de l'ENS - Parutions - série 4, 49 (2016)

Parutions < série 4, 49

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 49, fascicule 5 (2016)

Kari Astala, Daniel Faraco, Keith M. Rogers
Unbounded potential recovery in the plane
Annales scientifiques de l'ENS 49, fascicule 5 (2016), 1027-1051

Télécharger cet article : Fichier PDF

Résumé :
Reconstruction de potentiels non bornés dans le plan
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.

Mots-clefs : Problèmes inverses, théorie de la diffusion, convergence presque partout, Schrödinger.

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.

Keywords: Inverse problems, scattering, almost everywhere convergence, Schrödinger.

Class. math. : 35P25, 45Q05; 42B37, 35J10.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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