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.

翻译：本文引入计算策略，用于通过多种度量方式量化有界自伴算子谱的"尺寸"，包括勒贝格测度、分形维数、连通分量（或间隙）数量以及其他谱特征。我们的研究动机源于对几乎周期算子的研究，特别是作为准晶体模型出现的算子。这类算子以复杂的层次结构著称，常表现出精细的谱特性（如康托谱），这在量子力学系统和材料科学研究中具有重要意义。我们提出了一系列算法，可在不同假设下计算这些特性，并通过可解性复杂度指数（SCI）层次结构探讨其理论内涵。该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升实际场景中谱分析的精度提供了严格框架。例如，我们通过证明极限周期薛定谔算子类的特定下界（不可行性结果），证明了所提方法的最优性。我们通过对一维和二维非周期系统进行前沿计算来验证方法，有效捕捉了这些复杂的谱特征。该研究成果显著促进了谱理论中理论层面与计算层面的连接，为抽象数学概念与物理科学及工程实际应用之间的鸿沟搭建了桥梁。基于本工作，我们针对特定模型的谱特性提出了若干猜想和待解问题。