A semiclassical Birkhoff normal form for symplectic magnetic wells

Abstract

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 ${\hslash }^{1/2}$, and semiclassical Weyl asymptotics for this operator.