[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
@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 = {http://www.numdam.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 - http://www.numdam.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 http://www.numdam.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. http://www.numdam.org/articles/10.5802/aif.3451/
[1] Stability for multidimensional inverse spectral problem, Commun. Partial Differ. Equations, Volume 15 (1990) no. 5, pp. 711-736 | DOI | Zbl
[2] Über eine Frage der Eigenwerttheorie, Z. Phys., Volume 53 (1929), pp. 690-695 | DOI | Zbl
[3] On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl., Volume 5 (1997) no. 6, p. 487-480 | MR | Zbl
[4] An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, Volume 297 (1987), pp. 524-527
[5] To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Commun. Partial Differ. Equations, Volume 17 (1992), pp. 767-804 | DOI | MR | Zbl
[6] Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Probl., Volume 33 (2017), p. 055009 | DOI | Zbl
[7] Stability estimate for an inverse wave equation and a multidimensional Borg–Levinson theorem, J. Differ. Equations, Volume 247 (2009) no. 2, pp. 465-494 | DOI | MR | Zbl
[8] Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, Volume 5 (2011) no. 4, pp. 745-773 | DOI | MR | Zbl
[9] An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci., Volume 54 (2018), pp. 679-728 | DOI | Zbl
[10] Spectral Geometry : Direct and Inverse Problems, Lecture Notes in Mathematics, 1207, Springer, 1986
[11] Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe, Acta Math., Volume 78 (1946), pp. 1-96 | DOI | Zbl
[12] Determining coefficients in a class of heat equations via boundary measurements, SIAM J. Math. Anal., Volume 32 (2001) no. 5, pp. 963-986 | DOI | MR
[13] Determining two coefficients in elliptic operators via boundary spectral data: a uniqueness result, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8), Volume 7 (2004) no. 1, pp. 207-230 | MR | Zbl
[14] Une introduction aux problèmes inverses elliptiques et paraboliques, Mathématiques & Applications, 65, Springer, 2009 | DOI
[15] Stability for the multi-dimensional Borg–Levinson theorem with partial spectral data, Commun. Partial Differ. Equations, Volume 38 (2013) no. 3, pp. 455-476 | DOI | MR | Zbl
[16] Determining a magnetic Schrödinger operator from partial Cauchy data, Commun. Math. Phys., Volume 271 (2007) no. 2, pp. 467-488 | Zbl
[17] Limiting Carleman weights and anisotropic inverse problems, Invent. Math., Volume 178 (2009) no. 1, pp. 119-171 | DOI | MR | Zbl
[18] The Calderón problem in transversally anisotropic geometries, J. Eur. Math. Soc., Volume 18 (2016) no. 11, pp. 2579-2626 | DOI | Zbl
[19] The X-ray transform for a generic family of curves and weights, J. Geom. Anal., Volume 18 (2008) no. 1, pp. 89-108 | DOI | MR | Zbl
[20] On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 15 (1951), pp. 309-360 | MR
[21] Some remarks on the multi-dimensional Borg–Levinson theorem, J. Math. Kyoto Univ., Volume 31 (1991) no. 3, pp. 743-753 | MR | Zbl
[22] Riemannian Geometry and Geometric Analysis, Springer, 1995 | DOI
[23] Multidimensional inverse problem with incomplete boundary spectral data, Commun. Partial Differ. Equations, Volume 23 (1998), pp. 55-95 | MR | Zbl
[24] Inverse boundary spectral problems, Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, 2001 | DOI
[25] Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Probl., Volume 20 (2004) no. 2, pp. 419-436 | DOI | MR | Zbl
[26] Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, J. Math. Pures Appl., Volume 104 (2015) no. 6, pp. 1160-1189 | DOI | MR | Zbl
[27] A multidimensional Borg–Levinson theorem for magnetic Schrödinger operators with partial spectral data, J. Spectr. Theory, Volume 8 (2018) no. 1, pp. 235-269 | DOI | MR | Zbl
[28] Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide, Rev. Mat. Iberoam., Volume 36 (2020) no. 3, pp. 671-710 | DOI | MR | Zbl
[29] Recovery of non compactly supported coefficients of elliptic equations on an infinite waveguide, J. Inst. Math. Jussieu, Volume 19 (2020) no. 5, pp. 1573-1600 | DOI | MR | Zbl
[30] Simultaneous determination of coefficients, internal sources and an obstacle of a diffusion equation from a single measurement (2020) (https://arxiv.org/abs/2007.08947)
[31] Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets, J. Differ. Equations, Volume 267 (2019) no. 4, pp. 2210-2238 | DOI | MR | Zbl
[32] The uniqueness of inverse problems for a fractional equation with a single measurement, Math. Ann., Volume 380 (2021) no. 3-4, pp. 1465-1495 | DOI | MR | Zbl
[33] Application of the boundary control method to partial data Borg-Levinson inverse spectral problem, Math. Control Relat. Fields, Volume 9 (2019), pp. 289-312 | DOI | MR | Zbl
[34] Global uniqueness in an inverse problem for time-fractional diffusion equations, J. Differ. Equations, Volume 264 (2018) no. 2, pp. 1146-1170 | DOI | MR
[35] Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., Volume 51 (2019) no. 2, pp. 627-647 | DOI | Zbl
[36] Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, Inverse Probl. Imaging, Volume 14 (2020) no. 5, pp. 819-839 | DOI | Zbl
[37] Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Commun. Math. Phys., Volume 327 (2014) no. 3, pp. 993-1009 | DOI | Zbl
[38] An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Probl., Volume 26 (2010) no. 8, 085012, 19 pages | MR | Zbl
[39] Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., Volume 163 (2014) no. 6, pp. 1071-1103 | MR | Zbl
[40] The inverse Strum–Liouville problem, Mat. Tidsskr. B, Volume 1949 (1949), pp. 25-30 | Zbl
[41] Non homogeneous boundary value problems and applications. Vol. I, Grundlehren der Mathematischen Wissenschaften, 181, Springer, 1972
[42] An n-dimensional Borg–Levinson theorem, Commun. Math. Phys., Volume 115 (1988) no. 4, pp. 595-605 | DOI | MR | Zbl
[43] Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., Volume 303 (1995) no. 3, pp. 377-388 | DOI | Zbl
[44] Multidimensional Borg–Levinson theorems for unbounded potentials, Asymptotic Anal., Volume 110 (2018) no. 3-4, pp. 203-226 | DOI | MR | Zbl
[45] Inverse problems for nonsmooth first order perturbations of the Laplacian, Ph. D. Thesis, University of Helsinki (Finland) (2004)
[46] Borg-Levinson theorem for magnetic Schrödinger operator, Bull. Greek Math. Soc., Volume 57 (2010), pp. 321-332 | Zbl
[47] Integral Geometry of Tensor Fields, De Gruyter, 1994 | DOI
[48] Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., Volume 123 (2004) no. 3, pp. 445-467 | MR | Zbl
[49] Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., Volume 2005 (2005) no. 17, pp. 1047-1061 | DOI | MR | Zbl
[50] An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Am. Math. Soc., Volume 338 (1993) no. 2, pp. 953-969 | Zbl
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