Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on \(L^2({\mathbb {R}}^d)\). Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.

光谱测量出现在许多应用中，例如量子力学，信号处理，共振现象和流体稳定性分析。同样，光谱分解（分解为纯点，绝对连续和奇异的连续部分）通常表征相关的物理特性，例如量子系统的长期动力学。尽管在计算光谱方面取得了新的结​​果，但仍然没有能够计算无限维法线算符的光谱度量或光谱分解的通用方法。先前的工作集中在可以使用分析公式（或其扰动）的特定示例上，或在具有很多结构的算子类别上。因此，一般的计算问题主要是开放的。我们通过提供第一套通用算法来解决此问题，该算法可计算频谱度量和各种算子的分解。给定自伴或or算子的矩阵表示，以使得每列以已知的渐近速率在无穷大处衰减，我们展示了如何计算频谱测度和分解。我们将讨论这些方法如何允许对象（例如功能演算）的计算，以及它们如何推广到一大类偏微分算子，例如，提供演化PDE（例如线性Schrödinger方程）的解。我们展示了如何计算频谱测度和分解。我们将讨论这些方法如何允许对象（例如功能演算）的计算，以及它们如何推广到一大类偏微分算子，例如，提供演化PDE（例如线性Schrödinger方程）的解。我们展示了如何计算频谱测度和分解。我们将讨论这些方法如何允许对象（例如功能演算）的计算，以及它们如何推广到一大类偏微分算子，例如，提供演化PDE（例如线性Schrödinger方程）的解。\（L ^ 2（{\ mathbb {R}} ^ d）\）。无限维中的计算频谱问题导致了可解决复杂性指数（SCI）层次结构，该层次结构对计算问题的难度进行了分类。我们对SCI层次结构中的度量计算，度量分解，光谱类型，函数演算以及Radon-Nikodym衍生物进行分类。新算法在实线和单位圆上的正交多项式示例中被证明是有效的（例如，分别给出了Favard定理和Verblunsky定理的计算实现），并将其应用于两个维准晶体。