On the Bethe-Sommerfeld conjecture for the polyharmonic operator

Abstract

We consider in $L^2(ℝ^{d}),d≥2$, the perturbed polyharmonic operator $H=(−Δ)^{l}+V, l > 0$, with a function $V$ periodic with respect to a lattice in $ℝ^d$. 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.