Bethe-Sommerfeld conjecture for pseudodifferential perturbation

We consider a periodic pseudodifferential operator $H=(-\mathrm{\Delta }{)}^l+A$ ($l>0$) in ${\mathbb{R}}^d$ which satisfies the following conditions: (i) the symbol of $H$ is smooth in $x$, and (ii) the perturbation $A$ has order smaller than $2l-1$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line.