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 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 $$L^2({\mathbb {R}}^d)$$ L 2 ( 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.

The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem

“…Our proof techniques for fundamental barriers in Theorem 2 stem from the mathematics behind the solvability complexity index (SCI) hierarchy ( 33 , 69 – 78 ). The SCI hierarchy generalizes the fundamental problems of Smale ( 79 , 80 ) on existence of algorithms and work by McMullen ( 81 ) and Doyle and McMullen ( 82 ).…”

Bulk Localised Transport States in Infinite and Finite Quasicrystals via Magnetic Aperiodicity

“…In Ref. [86], it is shown how to compute the action of the resolvent with error control via the rectangular truncations P f (n) ( Ĥ−z)P n . Using this, we compute a smoothed approximation of E via convolution with a rational kernel K for smoothing parameter > 0.…”

“…The contour integral is computed using quadrature and approximations of the resolvent (H − z) −1 via rectangular truncations as above (see Ref. [86] for details and an example of fractional diffusion on a quasicrystal). In particular, the rectangular truncation of the Hamiltonian is chosen adaptively through a posteriori error bounds.…”

“…In the scenario of the current paper, this can be done through the rectangular truncations P f (n) ( Ĥ − z)P n and solving the resulting overdetermined linear system in the least squares sense. The residual converges to zero as n → ∞ and can be used to provide the needed error bounds through an adaptive selection of n (see [86,Th. 2.1]).…”

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