ON THE DISCRETENESS OF THE SPECTRUM OF THE LAPLACIAN ON NONCOMPACT RIEMANNIAN MANIFOLDS

Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold M are established in terms of the isocapacitary function of M. The relevant capacity takes a different form according to whether M has finite or infinite volume. Conditions involving the more standard isoperimetric function of M can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.