Dense Phenomena for Ergodic Schr\"odinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent

TL;DR

This paper studies ergodic Schrödinger operators with potentials sampled along orbits of a homeomorphism, showing generic spectral and dynamical properties like finite gaps and smoothness of key functions, and answering a question about cocycles.

Contribution

It demonstrates that for a dense set of sampling functions, the spectrum has finitely many gaps and the integrated density of states and Lyapunov exponent are smooth, advancing understanding of spectral properties.

Findings

Spectrum has finitely many gaps for generic sampling functions.

Integrated density of states is smooth for these operators.

Lyapunov exponent is smooth and positive, confirming conjectures.

Abstract

We consider Schr\"odinger operators in $\ell^2(\Z)$ whose potentials are defined via continuous sampling along the orbits of a homeomorphism on a compact metric space. We show that for each non-atomic ergodic measure $\mu$, there is a dense set of sampling functions such that the associated almost sure spectrum has finitely many gaps, the integrated density of states is smooth, and the Lyapunov exponent is smooth and positive. As a byproduct we answer a question of Walters about the existence of non-uniform $\SL(2,\R)$ cocycles in the affirmative.