Magnetic Wells in Dimension Three

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