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 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.
Highlights
The analysis and computation of spectral properties of operators form core parts of many branches of science and mathematics, arising in diverse fields such as differential and integral equations, orthogonal polynomials, quantum mechanics, statistical mechanics, integrable systems and optics [15,43,44,69,110,129,138]
Given a normal operator T, there is an associated projection-valued measure, which we denote by E T, whose existence is guaranteed by the spectral theorem and whose support is σ (T ) [86,87,113]
We show how to compute spectral measure decompositions, the functional calculus, the density of the absolutely continuous part of the measure (Radon–Nikodym derivative) and different types of spectra
Summary
The analysis and computation of spectral properties of operators form core parts of many branches of science and mathematics, arising in diverse fields such as differential and integral equations, orthogonal polynomials, quantum mechanics, statistical mechanics, integrable systems and optics [15,43,44,69,110,129,138]. We solve this problem by providing the first set of algorithms for the computation of spectral measures for a large class of self-adjoint and unitary operators, namely, those whose matrix columns decay at a known asymptotic rate This paper provides classifications of spectral problems associated with E T (such as decompositions of the measure and spectrum) in the SCI hierarchy, some of which can be computed in one limit. We provide algorithms for these problems, and one of the main tools used is the computation of the resolvent operator R(z, T ) := (T − z I )−1 with error control (Theorem 2.1) This is possible through appropriate rectangular truncations of the infinite-dimensional operator. This means that the algorithms constructed can be recursively implemented with inexact input and restrictions to arithmetic operations over the rationals (it is straightforward to implement them using interval arithmetic), whereas the impossibility results hold in any model of computation (such as the Turing or BSS models)
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