Sufficient conditions on the continuous spectrum for ergodic Schr\"odinger Operators

TL;DR

This paper establishes new sufficient conditions under which ergodic Schrödinger operators have purely continuous spectrum, broadening previous results by weakening assumptions and utilizing Gordon's lemma.

Contribution

It introduces weaker hypotheses involving topological repetition property to ensure purely continuous spectrum for ergodic Schrödinger operators, extending prior work.

Findings

Generic operators have purely continuous spectrum under certain density conditions.

The topological repetition property suffices for the spectral conclusion.

A proof of Gordon's lemma is provided for broader applicability.

Abstract

We study the spectral types of the families of discrete one-dimensional Schr\"odinger 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 $f:\Omega\rightarrow\mathbb{R}$ is a continuous function. We show that a generic operator $H_\omega\in \{H_\omega\}_{\omega\in\Omega}$ has purely continuous spectrum if $\{T^n\alpha\}_{n\geq0}$ 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.