Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals

TL;DR

The work develops a rigorous computational framework for quantifying the size of spectra of bounded self-adjoint operators, focusing on Lebesgue measure, fractal dimensions, and gaps, with applications to almost-periodic and quasicrystal models. By introducing two algorithmic paradigms, (S1) with spectral covers and (S2) with distance-to-spectrum functions, and by proving covering lemmas and SCI classifications, the authors deliver provably optimal methods and lower bounds, validated on one- and two-dimensional models including the almost Mathieu and Fibonacci Hamiltonians and Penrose tilings. The results clarify when single-limit versus multi-limit computations are necessary, establish lower bounds via limit-periodic constructions, and provide practical, error-controlled computations of Cantor-like spectra and their fractal structure in higher dimensions. The findings bridge theory and computation in spectral theory, enabling computer-assisted proofs and guiding open problems for two-dimensional quasicrystal spectra and beyond.

Abstract

We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and other spectral characteristics. Our motivation comes from the study of almost-periodic operators, particularly those that arise as models of quasicrystals. Such operators are known for intricate hierarchical patterns and often display delicate spectral properties, such as Cantor spectra, which are significant in studying quantum mechanical systems and materials science. We propose a series of algorithms that compute these properties under different assumptions and explore their theoretical implications through the Solvability Complexity Index (SCI) hierarchy. This approach provides a rigorous framework for understanding the computational feasibility of these problems, proving algorithmic optimality, and enhancing the precision of spectral analysis in practical settings. For example, we show that our methods are optimal by proving certain lower bounds (impossibility results) for the class of limit-periodic Schrödinger operators. We demonstrate our methods through state-of-the-art computations for aperiodic systems in one and two dimensions, effectively capturing these complex spectral characteristics. The results contribute significantly to connecting theoretical and computational aspects of spectral theory, offering insights that bridge the gap between abstract mathematical concepts and their practical applications in physical sciences and engineering. Based on our work, we conclude with conjectures and open problems regarding the spectral properties of specific models.

Figures (20)

• Figure 1: Left: A Hofstadter butterfly consisting of $\mathrm{Sp}_{+}(\alpha,1)$ (see \ref{['AM_union_spectrum']}) as $\alpha$ varies. The colorbar shows the value of $q$ (shown up to $q=51$), where $\alpha=p/q$ with $p$ and $q$ coprime. Right: The set $\mathrm{Sp}_{+}(377/610,\lambda)$ for various $\lambda$ and a rational convergent $377/610$ of $(\sqrt{5}-1)/2$.
• Figure 2: Results for the almost Mathieu operator with $\alpha=(\sqrt{5}-1)/2$. We have only shown the results for $\lambda\in[0,1]$ due to Aubry duality $\mathrm{Sp}(H_{\lambda,\alpha,\theta})=\lambda\,\mathrm{Sp}(H_{1/\lambda,\alpha,\theta})$. Left: Output of $\Pi_1^A$ algorithm for the Lebesgue measure of the spectrum. The convergence to $4|1-|\lambda||$ is evident. Middle: Output of the $\Sigma_1^A$ algorithm for the number of connected components of the spectrum. The divergence to $+\infty$ is evident. Right: Output of the $\Pi_1^A$ algorithm for the logarithmic capacity of the spectrum. The convergence to $1$ is evident.
• Figure 3: Covering numbers from the proof of \ref{['thm:box_counting']} for $\mathrm{Sp}_{+}(\alpha,1)$ for two different values of $\alpha$. The left plot shows a scaling of exponent $1/2$, while the right plot shows an apparent lack of scaling and possible values for the upper and lower box-counting dimensions. It appears that $\underline{\mathrm{dim}}_B(\mathrm{Sp}_{+}(C,1))\approx1/2$ and $\overline{\mathrm{dim}}_B(\mathrm{Sp}_{+}(C,1))\approx2/3$.
• Figure 4: Results for the Hausdorff dimension of $\mathrm{Sp}_{+}(\alpha,1)$ for two different values of $\alpha$. On the left, we see apparent convergence $\lim_{n_2\rightarrow\infty}\lim_{n_1\rightarrow\infty}\Gamma_{n_2,n_1}=1/2$. On the right, we see evidence that $\mathrm{dim}_{\mathrm{H}}(\mathrm{Sp}_{+}(C,1))<1/2$. Both cases show the monotonicity in the successive limits and the $\Sigma_2^A$ classification from \ref{['thm_Haus']}.
• Figure 5: Covers of the spectrum of the Fibonacci Hamiltonian, $\mathrm{Sp}(H_{\lambda})$, with coupling constant $\lambda=3/4$ (left) and $\lambda=5/4$ (right). For each level $k$, $\sigma_{\lambda,k}$ denotes the spectrum of the $k$th periodic approximation to Fibonacci potential. The union $\sigma_{\lambda,k} \cup \sigma_{\lambda,k+1}$ of two consecutive periodic approximations covers the spectrum, $\mathrm{Sp}(H_\lambda)$. The fine-scale that emerges for the higher levels of $k$ here hints at the Cantor nature of $\mathrm{Sp}(H_\lambda)$. For $\lambda=3/4$, the $k=8$ cover has 26 disjoint intervals, the smallest of which has width $9.96\times 10^{-3}$; the smallest gap is $5.01\times 10^{-4}$. The analogous cover for $\lambda=5/4$ has 36 disjoint intervals, the smallest of which has width $7.73\times 10^{-3}$; the smallest gap is $2.56\times 10^{-5}$. (Zoom in to see fine structure for the larger values of $k$.)

Theorems & Definitions (27)

• Theorem 3.1
• proof
• Example 3.1: Lebesgue measure of $\mathrm{Sp}(A)$
• Example 3.2: Number of connected components of $\mathrm{Sp}(A)$
• Example 3.3: Capacity of $\mathrm{Sp}(A)$
• Example 3.4: Decision problems
• Corollary 3.1
• Definition 3.1
• Theorem 3.2: Box-counting dimension can be computed in one limit
• proof