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

TL;DR

This paper proves the absolute continuity of the spectrum for a periodic Schr"odinger operator in layered and cylindrical geometries under certain integrability conditions on the potential.

Contribution

It establishes the absolute continuity of the spectrum for Schr"odinger operators in layers and cylinders with periodic potentials, extending previous results to less regular potentials.

Findings

Spectrum is absolutely continuous under specified conditions.

Results apply to potentials in certain local Lp spaces.

Extends known spectral properties to new geometric settings.

Abstract

We consider the Schr\"odinger operator $H = -\Delta + V$ 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 $V \in L_{p, \loc}$, where $p$ is a real number greater than $d/2$ in the case of a layer, and $p > \max (d/2, d-2)$ for the cylinder.