Johnson-Schwartzman Gap Labelling for Metric and Discrete Decorated Graphs

TL;DR

This paper extends Johnson-Schwartzman gap-labelling theorems to metric and discrete graphs from ergodic dynamical systems, analyzing spectral gaps and their labels in complex graph structures.

Contribution

It generalizes gap-labelling results beyond one-dimensional systems to graphs with cycles, introducing new spectral methods and analyzing gap closures.

Findings

Not all admissible gap labels correspond to open spectral gaps.

Graph geometry can cause gap closing independently of the underlying dynamics.

The results apply to graphs arising from uniquely ergodic one-dimensional systems.

Abstract

We study Schr\"odinger operators on metric and discrete decorated graphs. The values taken by the integrated density of states (IDS) on spectral gaps are called gap labels. A natural question is which gap labels can occur. We answer this for graphs arising from uniquely ergodic one-dimensional dynamical systems by proving Johnson-Schwartzman gap-labelling theorems in both the metric and discrete settings. Our results extend Johnson-Schwartzman gap labelling beyond the standard one-dimensional setting. Unlike in one dimension, these graphs may contain cycles, which prevent the use of Sturm oscillation theory and require different spectral methods. We also analyze discontinuities of the IDS for certain graph families and show that not every admissible label corresponds to an open spectral gap. This reveals a mechanism of gap closing driven by graph geometry rather than by the underlying dynamics.