A semiclassical Birkhoff normal form for constant-rank magnetic fields

TL;DR

This paper develops a semiclassical analysis of the magnetic Laplacian with a constant-rank magnetic field, constructing Birkhoff normal forms to describe eigenvalues near a well in the semiclassical limit.

Contribution

It introduces a novel method of constructing three successive Birkhoff normal forms for the magnetic Laplacian with a constant-rank magnetic field, enabling precise eigenvalue expansions.

Findings

Eigenvalues expanded in powers of ^{1/2}

Successful construction of normal forms for spectral analysis

Applicable to magnetic fields with a unique, non-degenerate well

Abstract

We consider the semiclassical magnetic Laplacian $\mathcal{L}_h$ on a Riemannian manifold, with a constant-rank and non-vanishing magnetic field $B$. Under the localization assumption that $B$ admits a unique and non-degenerate well, we construct three successive Birkhoff normal forms to describe the spectrum of $\mathcal{L}_h$ in the semiclassical limit $\hbar \rightarrow 0$. We deduce an expansion of all the eigenvalues under a threshold, in powers of $\hbar^{1/2}$.