Existence of the spectral gap for elliptic operators

LetM be a connected, noncompact, complete Riemannian manifold, consider the operatorL=Δ+∇V for someV∈C ²(M) with expV] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap ofL. 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→∞ supL _ϱ<0 while the spectral gap does not exist if o is a pole and lim_ϱ→∞ infL _ϱ≥0. Moreover, the elliptic operators onR ^d are also studied. Research supported in part by AvH Foundation, NSFC(19631060), Fok Ying-Tung Educational Foundation and Scientific Research Foundation for Returned Overseas Chinese Scholars.