Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 2 (1999), 395-407.

Existence of the spectral gap for elliptic operators

Feng-Yu Wang

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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.


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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Ark. Mat., Volume 37, Number 2 (1999), 395-407.

Received: 26 November 1997
Revised: 1 April 1998
First available in Project Euclid: 31 January 2017

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1999 © Institut Mittag-Leffler


Wang, Feng-Yu. Existence of the spectral gap for elliptic operators. Ark. Mat. 37 (1999), no. 2, 395--407. doi:10.1007/BF02412223.

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