Magnetic Wells in Dimension Three

Puits magnétiques en dimension trois

Bernard Helffer (1, 2) , Yuri Kordyukov (3) , Nicolas Raymond (4) , San Vu Ngoc (4)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 Institute of Mathematics, Russian Academy of Sciences, Ufa
4 IRMAR - Institut de Recherche Mathématique de Rennes

This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in 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 non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of the square root of the semiclassical parameter.

Mots clés: Magnetic fields; Birkhoff normal forms; microlocal analysis

Domaines: Mathématiques [math]

Bernard Helffer, Yuri Kordyukov, Nicolas Raymond, San Vu Ngoc. Magnetic Wells in Dimension Three. Analysis & PDE, 2016, 9 (7), pp.1575-1608. ⟨10.2140/apde.2016.9.1575⟩. ⟨hal-01149246⟩