  Under certain assumptions (including $d\ge 2)$ we prove that the spectrum of
a scalar operator in $\mathscr{L}^2(\mathbb{R}^d)$ \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, $\varepsilon=O(h^\varkappa)$,
$\varkappa>0$.
  Further, we consider generalizations.
