Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals

Matthew J. Colbrook, Mark Embree, Jake Fillman
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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 Schr\"odinger 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.
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量化光谱尺寸的最佳算法及其在准晶体中的应用
我们引入计算策略,利用各种度量,如勒贝格度量、分形维数、连通成分数(orgaps)和其他谱特征,来测量有界自相关算子谱的 "大小"。我们的研究动机来自对近周期算子的研究,特别是那些作为类晶体模型出现的算子。众所周知,这类算子具有错综复杂的层次模式,而且经常显示出微妙的光谱特性,例如康托尔光谱,这对研究量子力学系统和材料科学具有重要意义。我们提出了一系列在不同假设条件下计算这些性质的算法,并通过可解性复杂性指数(SCI)层次结构探讨其理论意义。这种方法为理解这些问题的计算可行性、证明算法最优性以及提高实际环境中光谱分析的精度提供了一个严谨的框架。例如,我们通过证明极限周期薛定谔算子类的某些下界(不可能性结果),证明我们的方法是最优的。我们通过对一维和二维非周期性系统的最新计算展示了我们的方法,有效地捕捉了这些复杂的谱特性。这些结果为连接谱理论的理论和计算方面做出了重大贡献,提供了弥合抽象数学概念与其在物理科学和工程学中的实际应用之间差距的见解。基于我们的工作,我们最后提出了关于特定模型频谱特性的猜想和开放性问题。
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