  We consider open manifolds which are interiors of a compact manifold with
boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric
near the boundary. We show that the essential spectrum of the Laplace operator
on functions vanishes under the presence of a magnetic field which does not
define an integral relative cohomology class. It follows that the essential
spectrum is not stable by perturbation even by a compactly supported magnetic
field. We also treat magnetic operators perturbed with electric fields. In the
same context we describe the essential spectrum of the $k$-form Laplacian. This
is shown to vanish precisely when the $k$ and $k-1$ de Rham cohomology groups
of the boundary vanish. In all the cases when we have pure-point spectrum we
give Weyl-type asymptotics for the eigenvalue-counting function. In the other
cases we describe the essential spectrum.
