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On the Spectrum of Sturmian Hamiltonians of Bounded Type in a Small Coupling Regime

Alexandro Luna

TL;DR

The paper proves that for irrational frequencies of bounded-type, the Hausdorff dimension of the spectrum of a Sturmian-disordered discrete Schrödinger operator approaches 1 as the coupling tends to 0. It achieves this by harnessing the trace map formalism and recasting the spectral problem as a non-stationary sequence of toral Anosov maps, then constructing dynamical rectangles and Cantor-set thickness estimates to bound the spectrum’s dimension from below. A key innovation is developing non-stationary stable manifolds and extending toral maps to ensure uniform control near cusps, enabling a thickness-based dimension bound that yields dim_H(σ_{λ,α}) → 1 as λ → 0 for bounded-type α. The results generalize prior work that required eventual periodicity in the continued fraction expansion and provide a framework potentially applicable to broader quasicrystal models and weak-coupling regimes. These insights illuminate transport properties in time-dependent Schrödinger dynamics and connect hyperbolic dynamics techniques to spectral questions in a quasicrystalline setting.

Abstract

We prove that the Hausdorff dimension of the spectrum of a discrete Schrödinger operator with Sturmian potential of bounded type tends to one as coupling tends to zero. The proof is based on the trace map formalism.

On the Spectrum of Sturmian Hamiltonians of Bounded Type in a Small Coupling Regime

TL;DR

The paper proves that for irrational frequencies of bounded-type, the Hausdorff dimension of the spectrum of a Sturmian-disordered discrete Schrödinger operator approaches 1 as the coupling tends to 0. It achieves this by harnessing the trace map formalism and recasting the spectral problem as a non-stationary sequence of toral Anosov maps, then constructing dynamical rectangles and Cantor-set thickness estimates to bound the spectrum’s dimension from below. A key innovation is developing non-stationary stable manifolds and extending toral maps to ensure uniform control near cusps, enabling a thickness-based dimension bound that yields dim_H(σ_{λ,α}) → 1 as λ → 0 for bounded-type α. The results generalize prior work that required eventual periodicity in the continued fraction expansion and provide a framework potentially applicable to broader quasicrystal models and weak-coupling regimes. These insights illuminate transport properties in time-dependent Schrödinger dynamics and connect hyperbolic dynamics techniques to spectral questions in a quasicrystalline setting.

Abstract

We prove that the Hausdorff dimension of the spectrum of a discrete Schrödinger operator with Sturmian potential of bounded type tends to one as coupling tends to zero. The proof is based on the trace map formalism.
Paper Structure (31 sections, 28 theorems, 119 equations, 10 figures)

This paper contains 31 sections, 28 theorems, 119 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background Material
  3. Approach and Outline of Paper
  4. Trace Map Dynamics as Anosov Diffeomorphisms of $\mathbb{T}^2$
  5. Torus Model for $\lambda=0$
  6. Torus Model for Small $\lambda>0$
  7. Dynamical Tools and Setup
  8. Boxes and Local Cones
  9. Dynamical Rectangles
  10. Geometric Properties of Dynamical Rectangles
  11. Main Estimates and Non-Stationary Trace Map Dynamics
  12. Outline of Proof of Theorem \ref{['intermediate theorem']}
  13. Development of Constants
  14. Choice of Neighborhood
  15. Extended Toral Maps
  16. ...and 16 more sections

Key Result

Theorem 1

If $\alpha\in(0,1)$ is irrational of bounded-type, then

Figures (10)

  • Figure 1: Surface $S_{\frac{1}{4}}$ and line $L_{\frac{1}{4}}$.
  • Figure 2: Depiction of $S_{\lambda}\setminus O_{\rho}$.
  • Figure 3: Illustration of Lemma\ref{['linear overflow']}
  • Figure 4: Illustration of local boxes.
  • Figure 5: Illustration of a local cone.
  • ...and 5 more figures

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • ...and 58 more