  We study the eigenvalues of the magnetic Schroedinger operator associated
with a magnetic potential A and a scalar potential q, on a compact Riemannian
manifold M, with Neumann boundary conditions if the boundary is not empty. We
obtain several bounds for the spectrum. Besides the dimension and the volume of
the manifold, the geometric quantity which plays an important role in these
estimates is the first eigenvalue of the Hodge-de Rham Laplacian acting on
co-exact 1-forms. In the 2-dimensional case, this is nothing but the first
positive eigenvalue of the Laplacian acting on functions. As for the dependence
of the bounds on the potentials, it brings into play the mean value of the
scalar potential q, the L^2-norm of the magnetic field B=dA, and the distance,
taken in L^2, between the harmonic component of A and the subspace of all
closed 1-forms whose cohomology class is integral (that is, having integral
flux around any loop). In particular, this distance is zero when the first
cohomology group is trivial.
