Computing Spectral Measures and Spectral Types

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