On the Bethe-Sommerfeld conjecture for the polyharmonic operator

摘要

We consider in L-2(R-d), d greater than or equal to 2, the perturbed polyharmonic operator H = (-Delta)(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 6l > d + 2, previously the finiteness of the number of gaps was known for 4l > d + 1. The proof is based on arithmetic properties of the lattice and elementary perturbation theory. [References: 20]