Sufficient conditions on the continuous spectrum for ergodic Schrödinger Operators

Abstract

We study the spectral types of the families of discrete one-dimensional Schrödinger operators $\{H_{\omega }{\}}_{\omega \in \Omega }$, where the potential of each $H_{\omega }$ is given by $V_{\omega }(n)=f(T^n\omega )$ for $n\in \mathbb{Z}$, $T$ is an ergodic homeomorphism on a compact space $\Omega$ and $f:\Omega \to \mathbb{R}$ is a continuous function. We show that a generic operator $H_{\omega }\in \{H_{\omega }{\}}_{\omega \in \Omega }$ has purely continuous spectrum if $\{T^n\alpha {\}}_{n\ge 0}$ is dense in $\Omega$ for a certain $\alpha \in \Omega$. We also show the former result assuming only that $\{\Omega ,T\}$ satisfies topological repetition property ($TRP$), a concept introduced by Boshernitzan and Damanik (arXiv:0708.1263v1). Theorems presented in this paper weaken the hypotheses of the cited research and allow us to reach the same conclusion as those authors. We also provide a proof of Gordon's lemma, which is the main tool used in this work.