Thin Spectra for Periodic and Ergodic Word Models
Jake Fillman, Michala N. Gradner, Hannah J. Hendricks
TL;DR
The paper develops a simple, scale-local criterion (gap-richness) for opening spectral gaps in periodic word models and uses it to prove that generic limit-periodic Schrödinger operators have Cantor spectra of zero Hausdorff dimension. By establishing that non-ahyperbolicity at a single scale implies gap-rich behavior at all scales, the authors derive a general mechanism for generating extremely thin spectra in both discrete and continuum settings; this yields new results such as Cantor spectra of dimension zero that persist under zero-insertions and small couplings. The approach applies across discrete and continuum Schrödinger operators and is demonstrated via explicit constructions (sieving, polymer methods, and their continuum analogues), bridging to ergodic and almost-periodic systems. The findings advance understanding of spectral geometry in limit-periodic and quasi-periodic models and offer a robust framework for constructing almost-periodic potentials with vanishing spectral dimension, with implications for singular continuous spectra and related dynamical properties.
Abstract
We establish a new and simple criterion that suffices to generate many spectral gaps for periodic word models. This leads to new examples of ergodic Schrödinger operators with Cantor spectra having zero Hausdorff dimension that simultaneously may have arbitrarily small supremum norm together with arbitrarily long runs on which the potential vanishes.