Generic Continuous Spectrum for Ergodic Schr"odinger Operators

TL;DR

This paper investigates the spectral properties of discrete Schrödinger operators with potentials generated by minimal dynamical systems, establishing generic absence of eigenvalues under certain repetition conditions.

Contribution

It introduces topological and metric repetition properties and proves that generic continuous sampling functions lead to operators with no eigenvalues, extending previous results to broader dynamical systems.

Findings

Operators have no eigenvalues in a topological or metric sense under certain conditions.

Applicable to shifts and skew-shifts on the torus.

Provides a unified framework for spectral analysis of ergodic Schrödinger operators.

Abstract

We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.