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Magnetic wells in
dimension three
Bernard Helffer, Yuri Kordyukov, Nicolas Raymond and San Vũ Ngọc |
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Vol. 9 (2016), No. 7, 1575–1608
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DOI: 10.2140/apde.2016.9.1575
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Abstract |
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This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in the presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms à la Birkhoff. As a consequence, when the magnetic field admits a unique and nondegenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional -pseudodifferential operator whose Weyl’s symbol admits an asymptotic expansion in powers of . |
Keywords
magnetic fields, Birkhoff normal forms, microlocal analysis
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Mathematical Subject Classification 2010
Primary: 81Q20, 35P15
Secondary: 37G05, 70H15
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Milestones
Received: 29 September 2015
Revised: 6 June 2016
Accepted: 9 July 2016
Published: 7 November 2016
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Authors |
| Bernard Helffer | |
| Département de Mathématiques Bât. 425 Université Paris-Sud et CNRS 91405 Orsay Cedex France |
Laboratoire de Mathématiques Jean
Leray Université de Nantes 2 rue de la Houssinière BP 92208 44322 Nantes Cedex 3 France |
| Yuri Kordyukov | |
| Institute of Mathematics Russian Academy of Sciences 112 Chernyshevsky str. 450008 Ufa Russia |
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| Nicolas Raymond | |
| Institut de Recherches Mathématiques
de Rennes (UMR 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France |
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| San Vũ Ngọc | |
| Institut de Recherches Mathématiques
de Rennes (UMR 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France |
Institut Universitaire de
France 1 rue Descartes 75231 Paris Cedex 5 France |
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