On the Bethe-Sommerfeld conjecture for the polyharmonic operator

We consider in L²(ℝᵈ),d≥2, the perturbed polyharmonic operator H=(−Δ)ˡ+V, l > 0, with a function V periodic with respect to a lattice in ℝᵈ. We prove that the number of gaps in the spectrum of H is finite if $6 l > d+2$. Previously the finiteness of the number of gaps was known for $4 l > d+1$. The proof is based on arithmetic properties of the lattice and elementary perturbation theory.