Existence of the spectral gap for elliptic operators

Abstract

LetM be a connected, noncompact, complete Riemannian manifold, consider the operator L=Δ+∇V for some V∈C²(M) with exp[V] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_ϱ→∞ sup L_ϱ<0 while the spectral gap does not exist if o is a pole and lim_ϱ→∞ inf L_ϱ≥0. Moreover, the elliptic operators on R^d are also studied.