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Spectral regularity and defects for the Kohmoto model

Siegfried Beckus, Jean Bellissard, Yannik Thomas

Abstract

We study the Kohmoto model including Sturmian Hamiltonians and the associated Kohmoto butterfly. We prove spectral estimates for the operators using Farey numbers. In addition, we determine the impurities at rational rotations leading to the spectral defects in the Kohmoto butterfly. Our results are similar to the ones obtained for the Almost-Mathieu operator and the associated Hofstadter butterfly.

Spectral regularity and defects for the Kohmoto model

Abstract

We study the Kohmoto model including Sturmian Hamiltonians and the associated Kohmoto butterfly. We prove spectral estimates for the operators using Farey numbers. In addition, we determine the impurities at rational rotations leading to the spectral defects in the Kohmoto butterfly. Our results are similar to the ones obtained for the Almost-Mathieu operator and the associated Hofstadter butterfly.

Paper Structure

This paper contains 11 sections, 31 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. A short introduction to mechanical words and Farey numbers
  3. The spaces of mechanical systems
  4. The Farey numbers
  5. Various representations of the mechanical systems
  6. The Farey space
  7. The dynamical representation
  8. The graph representation
  9. Defects
  10. Spectral applications
  11. Optimality in the Kohmoto model

Key Result

Theorem 1.2

For all $V\in{\mathbb R}$, the spectral map is Lipschitz-continuous. In particular, there is a $C=C(V) > 0$ such that

Figures (3)

  • Figure 1: A plot of the Kohmoto butterfly for $V>4$. The spectrum $\sigma(H_{r,V})$ at each $r\in[0,1]\cap{\mathbb Q}$ is split up into three points: the lower limit $r_-$, the upper limit $r_+$ and $r$, confer Proposition \ref{['prop-Defects']}. Proposition \ref{['prop:DiscreteSpectrum=q_points']} asserts that the upper and lower limiting spectra have exactly one additional (compared to $\sigma(H_{r,V})$) point in each bounded spectral gap plus one point in one of the unbounded spectral gaps of $\sigma(H_{r,V})$. This is demonstrated on the right hand side for $r=\frac{2}{3}$ and $s=\frac{1}{4}$. The red lines indicate the closing of the spectral gaps which is used to prove the optimality of the spectral estimates in Theorem \ref{['theo-SpeCon']}.
  • Figure 2: The building blocks of the interval-Farey tree are plotted.
  • Figure 3: The first levels of the Farey tree ${\mathcal{T}}_F$, its boundary $\partial {\mathcal{T}}_F$ and (in the gray box) the Farey space $\overline{[0,1]}_F$ are sketched. Each rational point is splits up into three points (except $0$ and $1$), confer Proposition \ref{['prop-convergence_in_farey_topo']}.

Theorems & Definitions (73)

  • Example 1.1
  • Theorem 1.2
  • proof
  • Corollary 1.3
  • proof
  • Proposition 2.1: BBdN20
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 63 more