A semiclassical Birkhoff normal form for constant-rank magnetic fields

Abstract

We consider the semiclassical magnetic Laplacian Lh 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 Lh in the semiclassical limit → 0. We deduce an expansion of all the eigenvalues under a threshold, in powers of 1/2.