Magnetic wells in dimension three

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 $\hslash$-pseudodifferential operator whose Weyl’s symbol admits an asymptotic expansion in powers of ${\hslash }^{\frac{1}{2}}$.