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 =d A 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 Lh = (ihd + A)∗(ihd + A). We construct a semiclassical Birkhoff normal form for Lh and deduce new asymptotic expansions of the smallest eigenvalues in powers of h1/2 in the limit h →0. In particular we see the influence of the kernel of B on the spectrum: it raises the energies at order h3/2.