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The Dry Ten Martini Problem for Sturmian Hamiltonians

Ram Band, Siegfried Beckus, Raphael Loewy

Abstract

The Dry Ten Martini Problem for Sturmian Hamiltonians is affirmatively solved. Concretely, we prove that all spectral gaps are open for Schrödinger operators with Sturmian potentials and non-vanishing coupling constant. A key approach towards the solution is a representation of the spectrum as the boundary of an infinite tree. This tree is constructed via particular periodic approximations and it encodes substantial spectral characteristics.

The Dry Ten Martini Problem for Sturmian Hamiltonians

Abstract

The Dry Ten Martini Problem for Sturmian Hamiltonians is affirmatively solved. Concretely, we prove that all spectral gaps are open for Schrödinger operators with Sturmian potentials and non-vanishing coupling constant. A key approach towards the solution is a representation of the spectrum as the boundary of an infinite tree. This tree is constructed via particular periodic approximations and it encodes substantial spectral characteristics.
Paper Structure (33 sections, 57 theorems, 317 equations, 17 figures)

This paper contains 33 sections, 57 theorems, 317 equations, 17 figures.

Table of Contents

  1. Introduction and main results
  2. The Dry Ten Martini Problem
  3. The spectra of the periodic (rational) approximations of $H_{\alpha,V}$
  4. The spectral $\alpha$-tree
  5. Connecting $\partial\mathcal{T}_{\alpha}$ with $\sigma(H_{\alpha,V})$ and proving Theorem \ref{['thm: all gaps are open']}
  6. A bird's-eye view on the spectrum of Sturmian Hamiltonians
  7. Sharpening the A-B type theorem and introducing fundamental concepts
  8. The space $\mathscr{C}$ of finite continued fraction expansions
  9. The spectra $\left\{ \sigma_{\textbf{c}}\right\} _{\textbf{c}\in\mathscr{C}}$
  10. Classification of spectral bands in $\sigma_{\textbf{c}}(V)$ to $A$ and $B$ types.
  11. Basic spectral analysis - preliminary tools for the proofs
  12. Lipschitz continuity of spectral band edges
  13. Perturbation argument
  14. Admissibility, index relations and the forward type properties
  15. Counting spectral bands and eigenvalues
  16. ...and 18 more sections

Key Result

Theorem 1.1

For all $\alpha\in[0,1]\setminus\mathbb{Q}$ and $V\in\mathbb{R}\setminus\{0\}$,

Figures (17)

  • Figure 1.1: (1) The root of the tree graph $T$ and two adjacent vertices. (2) A vertex $v$ in level $k$ (for $k\geq0$) and its outgoing edges to level $k+1$ and $k+2$.
  • Figure 1.2: An example of a spectral $\alpha$-tree is sketched if $\alpha$ has continued fraction expansion $\left(c_{k}\right)_{k=0}^{\infty}$ starting with $0,1,2,3$, see Definition \ref{['def: spectral approximants tree']}. The associated spectral bands are sketched as well. The two vertices $u,w$ which are marked satisfy $u\prec w$, but for their corresponding spectral bands, $\Psi(u)\not\prec\Psi(w)$ since $\left(\Psi(w)\right)(V)\subseteq_{\mathrm{strict}}\left(\Psi(u)\right)(V)$ for $V=1$.
  • Figure 2.1: A plot of various spectra for different $\textbf{c}\in\mathscr{C}$. The spectral bands are colored according to their backward types ($A$ in blue and $B$ in red). The embedding of these spectral bands within the Kohmoto butterfly is highlighted. The reader is referred to Example \ref{['exa: backward type =00005B0,0=00005D and =00005B0,0,1=00005D']}, Example \ref{['exa: duality of A-B types']} and Example \ref{['exa: Forward property']} for a detailed description.
  • Figure 4.1: A sketch for the proof of (c) and (d) in Lemma \ref{['lem: Floquet-Bloch matrices - n times fundamental domain']}.
  • Figure 4.2: A sketch for the statement of Lemma \ref{['lem: index identities for spectral bands']}. Note that if $I_{\textbf{c}}$ is of backward type $A$ for $V>4$, then there is a spectral band between $J_{[\textbf{c},m]}$ and $K_{[\textbf{c},m]}$. Otherwise, there is no spectral band between them, namely $\hbox{ind}(K_{[\textbf{c},m]})=\hbox{ind}(J_{[\textbf{c},m]})+1$. This is indicated by the question mark.
  • ...and 12 more figures

Theorems & Definitions (157)

  • Theorem 1.1: All gaps are there
  • Proposition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Remark
  • Definition 1.5
  • Remark
  • Definition 1.6
  • Theorem 1.7
  • Example 1.8
  • ...and 147 more