EXISTENCE OF THE SPECTRAL GAP FOR ELLIPTIC OPERATORS

FENG-YU WANG

Abstract. Let M be a connected, noncompact, complete Riemannian manifold, consider the operator L = Δ + ∇V for some V ∈ C^{2}(M) with exp[V] integrable w.r.t. 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 Lρ < 0 while the spectral gap does not exist if o is a pole and lim_{ρ→∞} inf Lρ ≥ 0. Moreover, the elliptic operators on ℝ^{d} are also studied.