  We describe the spectrum of the $k$-form Laplacian on conformally cusp
Riemannian manifolds. The essential spectrum is shown to vanish precisely when
the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. We give
Weyl-type asymptotics for the eigenvalue-counting function in the purely
discrete case. In the other case we analyze the essential spectrum via positive
commutator methods and establish a limiting absorption principle. This implies
the absence of the singular spectrum for a wide class of metrics. We also
exhibit a class of potentials $V$ such that the Schr\"odinger operator has
compact resolvent, although $V$ tends to $-\infty$ in most of the infinity. We
correct a statement from the literature regarding the essential spectrum of the
Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a
conjecture about the existence of such manifolds in dimension four whose cusps
are rational homology spheres.
