# Discrete Schrödinger operators with potentials defined by measurable sampling functions over Liouville torus rotations

### Alexander Y. Gordon

University of North Carolina at Charlotte, USA

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## Abstract

We consider a one-dimensional discrete Schrödinger operator $H_\omega=\Delta+v_\omega$ whose potential $v_\omega$ has the form $v_\omega(j)=V(\omega+j\bar{\alpha}), j\in\mathbb{Z}$. Here $\omega\in\mathbb{T}^k=\mathbb{R}^k/\mathbb{Z}^k,\ \alpha\in\mathbb{R}^k,\ \ \bar{\alpha}$ is the projection of $\alpha$ on $\mathbb{T}^k$, and the function $V\colon \mathbb{T}^k\to\mathbb{C}$ is Borel measurable. We show that if the frequency vector $\alpha$ is Liouville (the sequence $\{\nu \bar{\alpha}\}_{\nu\in\mathbb N}$ has a subsequence converging to $0$ fast enough), then for Lebesgue almost every $\omega\in\mathbb T^k$ the point spectrum of the operator $H_\omega$ is empty.

## Cite this article

Alexander Y. Gordon, Discrete Schrödinger operators with potentials defined by measurable sampling functions over Liouville torus rotations. J. Fractal Geom. 4 (2017), no. 4 pp. 329–337

DOI 10.4171/JFG/53