  We study the Schroedinger operator with a constant magnetic field in the
exterior of a compact domain in $\mathbb{R}^{2d}$, $d\geq 1$. The spectrum of
this operator consists of clusters of eigenvalues around the Landau levels. We
give asymptotic formulas for the rate of accumulation of eigenvalues in these
clusters. When the compact is a Reinhart domain we are able to show a more
precise asymptotic formula.
