Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale’s comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.

计算光谱是计算数学的核心问题，在整个科学领域有着丰富的应用。然而，在许多应用中，获得频谱的近似值是不够的。通常，确定光谱的几何特征（例如勒贝格测度、容量或分形维数、不同类型的光谱半径和数值范围）或检测基本光谱中的间隙以及有限截面方法的相应故障至关重要。尽管在计算谱方面取得了新的成果，并且人们对这些几何问题产生了浓厚的兴趣，但仍然没有能够计算无限维算子谱的此类几何特征的通用方法。我们提供了第一个算法来计算许多这些长期存在的问题（包括上述问题）。正如计算示例所示，新算法产生了一个新方法库。无限维计算谱问题的最新进展催生了可解复杂度指数（SCI）层次结构，它对计算问题的难度进行了分类。这些结果表明，无限维谱问题产生了复杂的无限分类理论，决定了可以解决哪些谱问题以及使用哪种类型的算法。这与 S. Smale 在 20 世纪 80 年代发起的计算数学基础综合项目密切相关。我们对 SCI 层次结构中光谱几何特征的计算进行分类，使我们能够精确确定计算机可以实现的边界（在任何计算模型中）并证明我们的算法是最优的。我们还提供了一种新的通用技术来建立 SCI 层次结构的下限，这既大大简化了以前的 SCI 论证，又允许新的、以前无法实现的分类。