Thin Spectra for Periodic and Ergodic Word Models

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.

💡 Research Summary

The paper introduces a remarkably simple yet powerful criterion that guarantees the creation of many spectral gaps for a broad class of one‑dimensional Schrödinger operators built from “word models.” The authors work with both discrete and continuum settings, focusing on potentials that are limit‑periodic (i.e., limits of periodic sequences) or more generally generated by ergodic subshifts over a finite alphabet.

Two central notions are defined: “gap‑rich” and “non‑hyperbolic.” A family of transfer‑matrix maps Φ : X × ℝ → SL(2,ℝ) is called gap‑rich (with a discrete exceptional set of energies) if for every energy outside that set, any finite word x can be perturbed by an arbitrarily small amount to a new word y such that the associated transfer matrix Φ(y,E) is hyperbolic. Hyperbolicity of the matrix corresponds exactly to the energy lying in a spectral gap. In contrast, non‑hyperbolicity is a much weaker, global condition: at the first (single‑letter) scale there exists at least one letter a∈X for which Φ(a,E) is hyperbolic.

The main abstract result (Theorem 2.3) shows that if Φ is analytic in the Banach‑space variable and is non‑hyperbolic, then it is automatically gap‑rich. The proof relies on analytic perturbation theory: the set of parameters for which a matrix fails to be hyperbolic is a proper analytic subvariety, hence its complement is dense and open. Consequently, checking non‑hyperbolicity at a single scale suffices to guarantee the ability to open gaps at all higher scales. This dramatically simplifies the verification compared with earlier approaches (e.g., the non‑commutation criteria of