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Spectral approximation for substitution systems

Ram Band, Siegfried Beckus, Felix Pogorzelski, Lior Tenenbaum

TL;DR

The paper develops a general framework to study spectral convergence of aperiodic Schrödinger operators defined by substitutions on lattices inside Lie groups with dilation. It introduces dilation and substitution data, substitution graphs, and testing tuples to obtain a verifiable criterion: spectral convergence of H_{S^n(ω_0)} to H_ω is equivalent to Hausdorff convergence of the orbit to the substitution subshift, detectable via finite-path properties on G_S(T,N_T). It further proves an exponential rate of convergence for the spectra in terms of the stretch factor λ_0, and provides both upper and lower bounds, including Lipschitz variants when coefficients are regular. The framework is illustrated through block substitutions and a non-abelian Heisenberg-group example, showing periodic approximations exist in some higher-dimensional settings and not in others, and offering an algorithm to minimize testing domains for practical computations. These results advance spectral analysis of higher-dimensional aperiodic order and enable efficient periodic approximations of a broad class of substitution-driven Schrödinger operators.

Abstract

We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $\mathbb{Z}^d$.

Spectral approximation for substitution systems

TL;DR

The paper develops a general framework to study spectral convergence of aperiodic Schrödinger operators defined by substitutions on lattices inside Lie groups with dilation. It introduces dilation and substitution data, substitution graphs, and testing tuples to obtain a verifiable criterion: spectral convergence of H_{S^n(ω_0)} to H_ω is equivalent to Hausdorff convergence of the orbit to the substitution subshift, detectable via finite-path properties on G_S(T,N_T). It further proves an exponential rate of convergence for the spectra in terms of the stretch factor λ_0, and provides both upper and lower bounds, including Lipschitz variants when coefficients are regular. The framework is illustrated through block substitutions and a non-abelian Heisenberg-group example, showing periodic approximations exist in some higher-dimensional settings and not in others, and offering an algorithm to minimize testing domains for practical computations. These results advance spectral analysis of higher-dimensional aperiodic order and enable efficient periodic approximations of a broad class of substitution-driven Schrödinger operators.

Abstract

We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over .
Paper Structure (19 sections, 32 theorems, 100 equations, 6 figures, 1 algorithm)

This paper contains 19 sections, 32 theorems, 100 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction and guiding example
  2. The underlying dynamical system
  3. A guiding example: The table tiling substitution
  4. Main results for general substitution systems
  5. Symbolic substitution systems
  6. Substitution graphs and testing domains
  7. First main result: Characterization of the convergence and its consequences
  8. Second main result: Exponential rate of convergence for the spectra
  9. Applications of the theory
  10. Block substitutions
  11. An example in the Heisenberg group
  12. Proof of the first main result
  13. Sufficient condition
  14. Proof of Theorem \ref{['thm:charact_convergence']}
  15. Proof of the second main result
  16. ...and 4 more sections

Key Result

Proposition 1.1

The following assertions are equivalent for the table tiling substitution and $\omega_0\in{\mathcal{A}}^{{\mathbb Z}^2}$ and $T:=\{0,1\}^2$. In particular, if $W(\omega_0)_T\subseteq W(S)$, then these equivalent conditions are satisfied.

Figures (6)

  • Figure 1: The first three iterations of the table tiling substitution starting from the letter $\textcolor{red}{\Huge\bullet}$.
  • Figure 2: The $S$-legal patches for the table tiling substitution with support $T=\{0,1\}^2$.
  • Figure 3: Subgraphs of the graph $G_{\textit{table}}$ are plotted.
  • Figure 4: The configurations $\omega_{rb},S(\omega_{rb}),\omega_{gy},S(\omega_{gy})\in{\mathcal{A}}^{{\mathbb Z}^2}$ are plotted. The gray shaded area indicated the block that is periodically extended.
  • Figure 5: A substitution rule where black bullet represents the letter $a$ and a gray bullet represents the letter $b$.
  • ...and 1 more figures

Theorems & Definitions (79)

  • Proposition 1.1
  • proof
  • Corollary 1.2
  • proof
  • Proposition 1.3
  • Remark 1.4
  • proof
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3: BHP21-Symbolic
  • ...and 69 more