A semiclassical Birkhoff normal form for constant-rank magnetic fields

This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field $B=dA$ has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian $H=|p-A(q){|}^2$ near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the semiexcited spectrum of the magnetic Laplacian ${\mathcal{ℒ}}_{\hslash }={(i\hslash d+A)}^{\ast }(i\hslash d+A)$. We construct a semiclassical Birkhoff normal form for ${\mathcal{ℒ}}_{\hslash }$ and deduce new asymptotic expansions of the smallest eigenvalues in powers of ${\hslash }^{1∕2}$ in the limit $\hslash \to 0$. In particular we see the influence of the kernel of $B$ on the spectrum: it raises the energies at order ${\hslash }^{3∕2}$.