  In this article we prove a generalization of Weyl's criterion for the
spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will
apply this new criterion in combination with Cheeger-Fukaya-Gromov and
Cheeger-Colding theory to study the $k$-form essential spectrum over a complete
manifold with vanishing curvature at infinity or asymptotically nonnegative
Ricci curvature.
  In addition, we will apply the generalized Weyl criterion to study the
variation of the spectrum of a self-adjoint operator under continuous
perturbations of the operator. In the particular case of the Laplacian on
$k$-forms over a complete manifold we will use these analytic tools to find
significantly stronger results for its spectrum including its behavior under a
continuous deformation of the metric of the manifold.
