Opening Gaps in the Spectrum of Strictly Ergodic Schr\"odinger Operators

TL;DR

This paper demonstrates that for Schr"odinger operators with dynamically defined potentials, all spectral gaps are generically open by continuously deforming sampling functions, using analysis of $SL(2,R)$ cocycles.

Contribution

It shows that collapsed spectral gaps can be opened through continuous deformations of sampling functions, ensuring generic operators have fully open spectra.

Findings

All spectral gaps are open for generic sampling functions.

Collapsed gaps can be opened via continuous deformations.

Analysis of $SL(2,R)$ cocycles is central to the proof.

Abstract

We consider Schr\"odinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap, the sampling function may be continuously deformed so that the gap immediately opens. As a corollary, we conclude that for generic sampling functions, all gaps are open. The proof is based on the analysis of continuous $SL(2,R)$ cocycles, for which we obtain dynamical results of independent interest.