A semiclassical Birkhoff normal form for symplectic magnetic wells

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schrödinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator