  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.
