Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

摘要

We consider a periodic self-adjoint pseudo-differential operator H=(-Δ)~m+B, m > 0, in ?~d which satisfies the following conditions: (i) the symbol of B is smooth in x, and (ii) the perturbation B has order less than 2m. Under these assumptions, we prove that the spectrum of H contains a half-line. This, in particular implies the Bethe-Sommerfeld conjecture for the Schr?dinger operator with a periodic magnetic potential in all dimensions.