Abstract.
We consider the elliptic operator P(D)+V in ℝd, 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 ∞(ℝd). 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 {ξℝd: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.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 10 October 2001 / Published online: 28 March 2003
Mathematics Subject Classification (1991): 35J10
Research supported in part by NSF Grant DMS-9732894.
Rights and permissions
About this article
Cite this article
Shen, Z. On the Bethe-Sommerfeld conjecture for higher-order elliptic operators. Math. Ann. 326, 19–41 (2003). https://doi.org/10.1007/s00208-003-0395-z
Issue Date:
DOI: https://doi.org/10.1007/s00208-003-0395-z