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What is the Wigner function closest to a given square integrable function?

2019-07-29Contribution to journal Restricted access
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dc.contributor.authorBen-Benjamin, J.S.
dc.contributor.authorCohen, J.
dc.contributor.authorDias, Nuno C.
dc.contributor.authorLoughlin, P.
dc.contributor.authorPrata, João N.
dc.date.accessioned2025-03-24T16:38:08Z
dc.date.available2025-03-24T16:38:08Z
dc.date.issued2019-07-29
dc.description.abstractWe consider an arbitrary square integrable function F on the phase-space and look for the Wigner function closest to it with respect to the L2 norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert--Schmidt operator withWeyl symbol F. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a bona fide Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results, we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.por
dc.description.versioninfo:eu-repo/semantics/publishedVersionpt_PT
dc.identifier.citationBen-Benjamin, J. S., Cohen, L., Dias, N. C., Loughlin, P., & Prata, J. N. (2018). What is the Wigner function closest to a given square integrable function?. SIAM Journal on Mathematical Analysis, 50(5), 5161-5197.
dc.identifier.doihttps://doi.org/10.1137/18M116633X
dc.identifier.urihttp://hdl.handle.net/10400.26/57408
dc.language.isoengpt_PT
dc.peerreviewedyes
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subjectWeyl quantization
dc.subjectWigner functions
dc.subjectWigner approximation
dc.subjectHilbert--Schmidt operators
dc.titleWhat is the Wigner function closest to a given square integrable function?pt_PT
dc.typecontribution to journal
dspace.entity.typePublication
oaire.citation.endPage5197
oaire.citation.issue8
oaire.citation.startPage5161
oaire.citation.titleSIAM Journal on Mathematical Analysis
oaire.citation.volume50
oaire.versionhttp://purl.org/coar/version/c_970fb48d4fbd8a85
rcaap.typeotherpt_PT

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