Absolute continuity of the spectrum of the periodic Scrödinger operator in a layer and in a smooth cylinder

The Schrödinger operator $H=\Delta+V$ in a layer or in a $d$-dimensional cylinder is considered. The function $V$ is suppored to be periodic with respect to some lattice. The absolute continuity of the spectrum of $H$ is established under the following conditions: $V\in L_{p,\mathrm{loc})}$ where $p>d/2$ in the case of a layer, and $p>>\max(d/2,d-2)$ in the case of a cylinder. Bibl. 14 titles.