Absolute continuity of the spectrum of the periodic Schr\"odinger operator in a layer and in a smooth cylinder

We consider the Schr"odinger operator $H=-\mathrm{\Delta }+V$ in a layer or in a $d$-dimensional cylinder. The potential $V$ is assumed to be periodic with respect to some lattice. We establish the absolute continuity of $H$, assuming $V\in L_{p,\text{loc}}$, where $p$ is a real number greater than $d\mathrm{/}2$ in the case of a layer, and $p>\max (d\mathrm{/}2,d-2)$ for the cylinder.