Magnetic Wells in Dimension Three

TL;DR

This paper analyzes the semiclassical behavior of the 3D magnetic Laplacian with magnetic confinement, identifying three scales and effective Hamiltonians, and reducing spectral analysis to a 1D pseudo-differential operator under certain conditions.

Contribution

It introduces a novel microlocal normal form approach to describe three semiclassical scales and simplifies spectral analysis to a 1D operator for magnetic wells.

Findings

Identifies three semiclassical scales in magnetic confinement.

Derives effective quantum Hamiltonians for each scale.

Reduces spectral analysis to a 1D pseudo-differential operator.

Abstract

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 $\hbar$-pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.