Vol. 9, No. 7, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Magnetic wells in dimension three

Bernard Helffer, Yuri Kordyukov, Nicolas Raymond and San Vũ Ngọc

Vol. 9 (2016), No. 7, 1575–1608
DOI: 10.2140/apde.2016.9.1575
Abstract

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 1 2 .

Keywords
magnetic fields, Birkhoff normal forms, microlocal analysis
Mathematical Subject Classification 2010
Primary: 81Q20, 35P15
Secondary: 37G05, 70H15
Milestones
Received: 29 September 2015
Revised: 6 June 2016
Accepted: 9 July 2016
Published: 7 November 2016
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
Nicolas Raymond
Institut de Recherches Mathématiques de Rennes (UMR 6625)
Université de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex
France
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