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Mathematische Annalen

, Volume 326, Issue 1, pp 19–41 | Cite as

On the Bethe-Sommerfeld conjecture for higher-order elliptic operators

  • Zhongwei Shen

Abstract.

 We consider the elliptic operator P(D)+V in ℝ d , d≥2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2ℓ with a homogeneous convex symbol P(ξ), and V is a real periodic function in L(ℝ d ). We show that the number of gaps in the spectrum of P(D)+V is finite if 4ℓ>d+1. If in addition, V is smooth and the convex hypersurface {ξℝ d :P(ξ)=1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D)+V is finite, provided 8ℓ>d+3 and 9≥d≥2, or 4ℓ>d−3 and d≥10.

Keywords

Periodic Function Gaussian Curvature Elliptic Operator Constant Coefficient Convex Hypersurface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Zhongwei Shen
    • 1
  1. 1.Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA (e-mail: shenz@ms.uky.edu)US

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