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dc.contributor.authorColbrook, Matthew J.
dc.date.accessioned2021-05-04T15:33:42Z
dc.date.available2021-05-04T15:33:42Z
dc.date.issued2021-04-11
dc.date.submitted2020-02-26
dc.identifier.issn0010-3616
dc.identifier.others00220-021-04072-4
dc.identifier.other4072
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/321929
dc.description.abstractAbstract: 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 L2(Rd). 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.
dc.languageen
dc.publisherSpringer Berlin Heidelberg
dc.subjectArticle
dc.titleComputing Spectral Measures and Spectral Types
dc.typeArticle
dc.date.updated2021-05-04T15:33:41Z
prism.endingPage501
prism.issueIdentifier1
prism.publicationNameCommunications in Mathematical Physics
prism.startingPage433
prism.volume384
dc.identifier.doi10.17863/CAM.69387
dcterms.dateAccepted2021-03-16
rioxxterms.versionofrecord10.1007/s00220-021-04072-4
rioxxterms.versionVoR
rioxxterms.licenseref.urihttp://creativecommons.org/licenses/by/4.0/
dc.contributor.orcidColbrook, Matthew J. [0000-0003-4964-9575]
dc.identifier.eissn1432-0916
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/L016516/1)


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