Bethe-Sommerfeld conjecture in semiclassical settings

Abstract

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

19 pp, 1 fig