A semiclassical Birkhoff normal form for constant-rank magnetic fields

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 $\mathrm{\hslash }\to 0$. We deduce an expansion of all the eigenvalues under a threshold, in powers of ${\mathrm{\hslash }}^{1\mathrm{/}2}$.