A Borg–Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold
Annales de l'Institut Fourier, Online first, 47 p.

We establish uniqueness and stability results for the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from the knowledge of boundary spectral data of the corresponding magnetic Schrödinger operator with Dirichlet boundary condition. The spectral data consist in the knowledge of asymptotic properties, that we specify hereafter, of the sequence of eigenvalues and Neumann traces of the corresponding sequence of eigenfunctions. We also prove similar results for Schrödinger operators with Neumann boundary conditions. To our knowledge our results are the first ones involving such weak boundary spectral data.

Nous établissons des résultats d’unicité et de stabilité pour le problème qui consiste à reconstruire, à partir de données spectrales au bord, le champ magnétique et le potentiel électrique, qui apparaissent dans une équation de Schrödinger magnétique sur une variété riemannienne compacte, avec une condition aux limites de Dirichlet. Les données spectrales consistent en la connaissance du comportement asymptotique, dans un sens que nous préciserons, de la suite des valeurs propres, de l’opérateur de Schrödinger magnétique avec une condition aux limites de Dirichlet, et des traces des dérivées normales des fonctions propres associées. Nous démontrons également des résultats similaires pour un opérateur de Schrödinger magnétique avec une condition aux limites de Neumann. A notre connaissance nos résultats sont les premiers concernant les problèmes spectraux inverses avec des données spectrales au bord aussi faibles.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3451
Classification: 35R30,  35J10,  35P99
Keywords: Borg–Levinson type theorem, magnetic Schrödinger operator, simple Riemannian manifold, uniqueness, stability estimate.
Bellassoued, Mourad 1; Choulli, Mourad 2; Dos Santos Ferreira, David 3; Kian, Yavar 4; Stefanov, Plamen 5

1 Université de Tunis El Manar Ecole Nationale d’Ingénieurs de Tunis LAMSIN, BP 37 1002 Tunis Le Belvédère (Tunisia)
2 Université de Lorraine 34 cours Léopold 54052 Nancy cedex (France)
3 Institut Élie Cartan de Lorraine, UMR CNRS 7502, équipe SPHINX, INRIA Université de Lorraine F-54506 Vandoeuvre-lès-Nancy Cedex (France)
4 Aix Marseille Univ, Université de Toulon CNRS, CPT Marseille (France)
5 Department of Mathematics Purdue University West Lafayette, IN 47907 (USA)
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Bellassoued, Mourad; Choulli, Mourad; Dos Santos Ferreira, David; Kian, Yavar; Stefanov, Plamen. A Borg–Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold. Annales de l'Institut Fourier, Online first, 47 p.

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