Absolute continuity of the periodic Schrödinger operator in transversal geometry

Abstract

We show that the spectrum of a Schrödinger operator on ${\mathbb{R}}^n,n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric potential in $L_{\mathrm{loc}}^{n=2}({\mathbb{R}}^n)$, is purely absolutely continuous. Previous results in the case of a general metric are obtained in [12], see also [9], under the assumption that the metric, as well as the potential, are reflection symmetric.