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

We consider the elliptic operator P(D)+ V in R , d ≥ 2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2 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 4 > d + 1. If in addition, V is smooth and the convex hypersurface {ξ ∈ R : P(ξ) = 1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D)+V is finite, provided 8 > d + 3 and 9 ≥ d ≥ 2, or 4 > d − 3 and d ≥ 10. Mathematics Subject Classification (1991): 35J10