no. 3
Semiclassical spectral gaps of the 3D Neumann Laplacian with constant magnetic field
[Trou spectral semiclassique pour le Laplacien Neumann 3D avec un champ magnétique constant]
Hérau, Frédéric1 ; Raymond, Nicolas2
1 Nantes Université, CNRS, LMJL 2 rue de la Houssinière, BP 92208 44322 Nantes cedex 3 (France)
2 Univ Angers, CNRS, LAREMA, SFR MATHSTIC 49000 Angers (France)
Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 915-972.

Cet article traite de l’analyse spectrale semiclassique du Laplacien magnétique sur un ouvert borné et régulier en dimension trois. Lorsque le champ magnétique est constant, nous établissons un développement asymptotique à cinq termes des plus petites valeurs propres. Ce dernier met en jeu une quantité géométrique définie le long du contour apparent de Ω dans la direction du champ. En particulier, nous prouvons la simplicité des valeurs propres.

This article deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a five-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of Ω in the direction of the field. In particular, we prove that they are simple.

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DOI : 10.5802/aif.3631
Classification : 35PXX, 81Q10, 81Q20
Keywords: magnetic Schrödinger operator, semiclassical analysis, eigenvalues
Mot clés : opérateur de Schrödinger magnétique, analyse semiclassique, valeurs propres

Hérau, Frédéric 1 ; Raymond, Nicolas 2

1 Nantes Université, CNRS, LMJL 2 rue de la Houssinière, BP 92208 44322 Nantes cedex 3 (France)
2 Univ Angers, CNRS, LAREMA, SFR MATHSTIC 49000 Angers (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de l'Institut Fourier},
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Hérau, Frédéric; Raymond, Nicolas. Semiclassical spectral gaps of the 3D Neumann Laplacian with constant magnetic field. Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 915-972. doi : 10.5802/aif.3631. https://aif.centre-mersenne.org/articles/10.5802/aif.3631/

[1] Bonnaillie-Noël, Virginie; Hérau, Frédéric; Raymond, Nicolas Magnetic WKB constructions, Arch. Ration. Mech. Anal., Volume 221 (2016) no. 2, pp. 817-891 | DOI | MR | Zbl

[2] Bonnaillie-Noël, Virginie; Hérau, Frédéric; Raymond, Nicolas Purely magnetic tunneling effect in two dimensions, Invent. Math., Volume 227 (2022) no. 2, pp. 745-793 | DOI | MR | Zbl

[3] Bony, Jean-Michel Sur l’inégalité de Fefferman–Phong, Sémin. Équ. Dériv. Partielles, Volume 1998-1999 (1999), III, 16 pages | Numdam | MR | Zbl

[4] Dauge, Monique; Helffer, Bernard Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators, J. Differ. Equations, Volume 104 (1993) no. 2, pp. 243-262 | DOI | MR | Zbl

[5] Fermanian Kammerer, Clotilde Opérateurs pseudo-différentiels semi-classiques, Chaos en mécanique quantique, Éditions de l’École polytechnique, 2014, pp. 53-100 | MR | Zbl

[6] Fournais, Soeren; Helffer, Bernard Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier, Volume 56 (2006) no. 1, pp. 1-67 | DOI | Numdam | MR | Zbl

[7] Fournais, Soeren; Helffer, Bernard Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, 77, Birkhäuser, 2010, xx+324 pages | DOI | MR

[8] Fournais, Soeren; Persson, Mikael Strong diamagnetism for the ball in three dimensions, Asymptotic Anal., Volume 72 (2011) no. 1-2, pp. 77-123 | DOI | MR | Zbl

[9] Fournais, Soeren; Sundqvist, Mikael Persson A uniqueness theorem for higher order anharmonic oscillators, J. Spectr. Theory, Volume 5 (2015) no. 2, pp. 235-249 | DOI | MR | Zbl

[10] Helffer, Bernard The Montgomery model revisited, Colloq. Math., Volume 118 (2010) no. 2, pp. 391-400 | DOI | MR | Zbl

[11] Helffer, Bernard; Kordyukov, Yuri; Raymond, Nicolas; Vũ Ngọc, San Magnetic wells in dimension three, Anal. PDE, Volume 9 (2016) no. 7, pp. 1575-1608 | DOI | MR | Zbl

[12] Helffer, Bernard; Mohamed, Abderemane Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., Volume 138 (1996) no. 1, pp. 40-81 | DOI | MR | Zbl

[13] Helffer, Bernard; Morame, Abderemane Magnetic bottles in connection with superconductivity, J. Funct. Anal., Volume 185 (2001) no. 2, pp. 604-680 | DOI | MR | Zbl

[14] Helffer, Bernard; Morame, Abderemane Magnetic bottles for the Neumann problem: the case of dimension 3, Proc. Indian Acad. Sci., Math. Sci., Volume 112 (2002) no. 1, pp. 71-84 Spectral and inverse spectral theory (Goa, 2000) | DOI | MR | Zbl

[15] Helffer, Bernard; Morame, Abderemane Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case), Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 105-170 | DOI | Numdam | MR | Zbl

[16] Keraval, Pierig Formules de Weyl par réduction de dimension. Applications à des Laplaciens électro-magnétiques, Ph. D. Thesis, Université de Rennes 1 (2018)

[17] Martinez, André A general effective Hamiltonian method, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 18 (2007) no. 3, pp. 269-277 | DOI | MR | Zbl

[18] Morin, Léo A semiclassical Birkhoff normal form for constant-rank magnetic fields (2019) | arXiv

[19] Morin, Léo A semiclassical Birkhoff normal form for symplectic magnetic wells, J. Spectr. Theory, Volume 12 (2022) no. 2, pp. 459-496 | DOI | MR | Zbl

[20] Raymond, Nicolas Bound states of the magnetic Schrödinger operator, EMS Tracts in Mathematics, 27, European Mathematical Society, 2017, xiv+380 pages | DOI | MR

[21] Sjöstrand, Johannes Semi-excited states in nondegenerate potential wells, Asymptotic Anal., Volume 6 (1992) no. 1, pp. 29-43 | DOI | MR | Zbl

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