Computing Spectral Measures and Spectral Types

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.