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
.
Keywords
magnetic fields, Birkhoff normal forms, microlocal analysis