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 limϱ→∞ sup Lϱ<0 while the spectral gap does not exist if o is a pole and limϱ→∞ inf Lϱ≥0. Moreover, the elliptic operators on Rd are also studied.
Funding Statement
Research supported in part by AvH Foundation, NSFC(19631060), Fok Ying-Tung Educational Foundation and Scientific Research Foundation for Returned Overseas Chinese Scholars.
Citation
Feng-Yu Wang. "Existence of the spectral gap for elliptic operators." Ark. Mat. 37 (2) 395 - 407, October 1999. https://doi.org/10.1007/BF02412223
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