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We consider a periodic self-adjoint pseudo-differential operator$H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions:(i) the symbol of $B$ is smooth in $\bx$, and (ii) the perturbation $B$ hasorder less than $2m$. Under these assumptions, we prove that the spectrum of$H$ contains a half-line. This, in particular implies the Bethe-SommerfeldConjecture for the Schr\"odinger operator with a periodic magnetic potential inall dimensions.

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations