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

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 $V:{\mathbb{T}}^k\to \mathbb{C}$ 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.