[Un théorème de Borg–Levinson pour les opérateurs de Schrödinger magnétiques sur une variété riemannienne]
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.
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.
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DOI : 10.5802/aif.3451
Keywords: Borg–Levinson type theorem, magnetic Schrödinger operator, simple Riemannian manifold, uniqueness, stability estimate.
Mot clés : théorème de Borg–Levinson, opérateur de Schrödinger magnétique, variété riemannienne simple, estimation de stabilité
Bellassoued, Mourad 1 ; Choulli, Mourad 2 ; Dos Santos Ferreira, David 3 ; Kian, Yavar 4 ; Stefanov, Plamen 5
CC-BY-ND 4.0
@article{AIF_2021__71_6_2471_0,
author = {Bellassoued, Mourad and Choulli, Mourad and Dos Santos Ferreira, David and Kian, Yavar and Stefanov, Plamen},
title = {A {Borg{\textendash}Levinson} theorem for magnetic {Schr\"odinger} operators on a {Riemannian} manifold},
journal = {Annales de l'Institut Fourier},
pages = {2471--2517},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {71},
number = {6},
year = {2021},
doi = {10.5802/aif.3451},
zbl = {07554452},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3451/}
}
TY - JOUR AU - Bellassoued, Mourad AU - Choulli, Mourad AU - Dos Santos Ferreira, David AU - Kian, Yavar AU - Stefanov, Plamen TI - A Borg–Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold JO - Annales de l'Institut Fourier PY - 2021 SP - 2471 EP - 2517 VL - 71 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3451/ DO - 10.5802/aif.3451 LA - en ID - AIF_2021__71_6_2471_0 ER -
%0 Journal Article %A Bellassoued, Mourad %A Choulli, Mourad %A Dos Santos Ferreira, David %A Kian, Yavar %A Stefanov, Plamen %T A Borg–Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold %J Annales de l'Institut Fourier %D 2021 %P 2471-2517 %V 71 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3451/ %R 10.5802/aif.3451 %G en %F AIF_2021__71_6_2471_0
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, Tome 71 (2021) no. 6, pp. 2471-2517. doi : 10.5802/aif.3451. https://aif.centre-mersenne.org/articles/10.5802/aif.3451/
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