Absolute continuity of the periodic Schr\"odinger operator in transversal geometry

Katsiaryna Krupchyk; Gunther Uhlmann

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arXiv · 1312.2989

Absolute continuity of the periodic Schr\"odinger operator in transversal geometry

Katsiaryna Krupchyk·Gunther Uhlmann

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Abstract

We show that the spectrum of a Schr\"odinger 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 $L^{n/2}_{\text{loc}}(\mathbb{R}^n)$, is purely absolutely continuous. Previously known results in the case of a general metric are obtained in [12], see also [8], under the assumption that the metric, as well as the potential, are reflection symmetric.

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Katsiaryna Krupchyk, Gunther Uhlmann. 2013-12-10. Absolute continuity of the periodic Schr\"odinger operator in transversal geometry. https://arxiv.org/abs/1312.2989

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