On the bethe-sommerfeld conjecture for higher-order elliptic operators

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 hyper-surface {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.