The uncertainty principle lemma under gravity and the discrete spectrum of Schr\"odinger operators

The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schr"odinger operator has a finite or infinite number of the discrete pectrum. In this paper, we will give a generalization of this lemma on Euclidean spaces to that on large classes of complete noncompact manifolds. Replacing Euclidean spaces by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some criterions for the above-type question.