Bethe-Sommerfeld Conjecture in Semiclassical Settings

Under certain assumptions (including $$d\ge 2)$$ we prove that the spectrum of a scalar operator in $$\mathscr {L}^2({\mathbb {R}}^d)$$ $$\begin{aligned} A_\varepsilon (x, hD)= A^0(hD) + \varepsilon B(x, hD), \end{aligned}$$ covers interval $$(\tau -\epsilon ,\tau +\epsilon )$$ , where $$A^0$$ is an elliptic operator and B(x, hD) is a periodic perturbation, $$\varepsilon =O(h^\varkappa )$$ , $$\varkappa >0$$ . Further, we consider generalizations.