Geometry and Spectrum in 2D Magnetic Wells

TL;DR

This paper analyzes the classical and spectral properties of a 2D magnetic Hamiltonian, using Birkhoff normal form to reduce the problem to one-dimensional Hamiltonians, extending recent high-energy spectral results.

Contribution

It introduces a method to study 2D magnetic Hamiltonians via Birkhoff normal form, enabling extension of spectral results to higher energies.

Findings

Dynamics and spectral theory reduced to 1D Hamiltonians

Extension of Helffer-Kordyukov results to higher energies

Spectral analysis facilitated by normal form techniques

Abstract

This paper is devoted to the classical mechanics and spectral analysis of a pure magnetic Hamiltonian in $\R^2$. It is established that both the dynamics and the semiclassical spectral theory can be treated through a Birkhoff normal form and reduced to the study of a family of one dimensional Hamiltonians. As a corollary, recent results by Helffer-Kordyukov are extended to higher energies.