Abstract
LetM be a connected, noncompact, complete Riemannian manifold, consider the operatorL=Δ+∇V for someV∈C 2(M) with exp[V] 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 limϱ→∞ supL ϱ<0 while the spectral gap does not exist if o is a pole and limϱ→∞ infL ϱ≥0. Moreover, the elliptic operators onR d are also studied.
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Research supported in part by AvH Foundation, NSFC(19631060), Fok Ying-Tung Educational Foundation and Scientific Research Foundation for Returned Overseas Chinese Scholars.
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Wang, FY. Existence of the spectral gap for elliptic operators. Ark. Mat. 37, 395–407 (1999). https://doi.org/10.1007/BF02412223
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DOI: https://doi.org/10.1007/BF02412223