Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonances, and fluid stability. Similarly, spectral decompositions (pure point, absolutely continuous and singular continuous) often characterise relevant physical properties such as 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 focus on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators with 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 Schr\"odinger equations on $L^2(\mathbb{R}^d)$. Computational spectral problems in infinite dimensions have led to the SCI hierarchy, which classifies the difficulty of computational problems. We classify 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 OPs on the real line and the unit circle (e.g. giving computational realisations of Favard's theorem and Verblunsky's theorem), and are applied to evolution equations on a 2D quasicrystal.
翻译:量子力学、 信号处理、 共振和流体稳定性等多种应用中都会出现光谱测量。 同样, 光谱分解( 纯点、 绝对连续和单单连续) 通常会描述量子系统的长期动态等相关物理属性。 尽管计算光谱有了新的结果, 但仍没有一般方法能够计算光度测量或无穷度正常操作员的光谱分解。 以往的努力侧重于分析公式( 或对其的扰动) 的具体实例, 或结构繁多的操作员类别。 因此一般的计算问题大多是公开的。 我们通过提供第一组通用的计算算法( 纯点、 绝对连续的和单向级的运算值) 来解决这个问题。 由于一个自合或单一操作员的矩阵表示, 每列的光谱度或光谱正常操作员分解分解。 我们展示了如何将光谱测量和分解的分解方法用于计算诸如功能计算的量值等物体, 以及它们如何将精度递解的计算结果转换成一个大级的运算法, 将S- dal dal dal dealdaldalation 的运算算算算算算算算算算算算算成成成为一个大等级, 。 在S 度上, 在变变变变变变变法中, 度上, 将S daldaldaldaldald 度的算法的计算法的计算法的数值的计算方法如何为一个大的计算法的计算法的算算算算算算算算算算算算算算算算法, 。