The lower bound of the Ricci curvature that yields the infinite number of the discrete spectrum of the Laplacian

Abstract

This paper discusses the question whether the discrete spectrum of the Laplace-Beltrami operator is infinite or finite. The borderline-behavior of the curvatures for this problem will be completely determined.

arXiv comment8 pages; add some contexts in Introduction for readers, typos; typos, again; modified Abstract; Main theorem is generalized into more refined form and some applications are added in the last section. The journal office, which accepts this paper, strongly encourages the author to submit (or update) the final version of this paper to arXiv

References in corpus (1)

• The uncertainty principle lemma under gravity and the discrete spectrum of Schrödinger operators

Cited by in corpus (1)

• Positive and negative growth rates of measures and pointwise bounds of solutions of eigenvalue equations