Strong semiclassical approximation of Wigner functions for the Hartree dynamics

Handle URI:
http://hdl.handle.net/10754/599749
Title:
Strong semiclassical approximation of Wigner functions for the Hartree dynamics
Authors:
Athanassoulis, Agissilaos; Paul, Thierry; Pezzotti, Federica; Pulvirenti, Mario
Abstract:
We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit h → 0. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L 2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L 2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which - as it is well known - is not pointwise positive in general.
Citation:
Athanassoulis A, Paul T, Pezzotti F, Pulvirenti M (2011) Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Atti Accad Naz Lincei Cl Sci Fis Mat Natur: 525–552. Available: http://dx.doi.org/10.4171/RLM/613.
Publisher:
European Mathematical Publishing House
Journal:
Rendiconti Lincei - Matematica e Applicazioni
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
2011
DOI:
10.4171/RLM/613
Type:
Article
ISSN:
1120-6330
Sponsors:
A. Athanassoulis would like to thank the CMLS, Ecole polytechnique for its hospitality during the preparation of this work. He was also partially supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). F. Pezzotti was partially supported by Project CBDif-Fr ANR-08-BLAN-0333-01.
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Full metadata record

DC FieldValue Language
dc.contributor.authorAthanassoulis, Agissilaosen
dc.contributor.authorPaul, Thierryen
dc.contributor.authorPezzotti, Federicaen
dc.contributor.authorPulvirenti, Marioen
dc.date.accessioned2016-02-28T06:08:56Zen
dc.date.available2016-02-28T06:08:56Zen
dc.date.issued2011en
dc.identifier.citationAthanassoulis A, Paul T, Pezzotti F, Pulvirenti M (2011) Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Atti Accad Naz Lincei Cl Sci Fis Mat Natur: 525–552. Available: http://dx.doi.org/10.4171/RLM/613.en
dc.identifier.issn1120-6330en
dc.identifier.doi10.4171/RLM/613en
dc.identifier.urihttp://hdl.handle.net/10754/599749en
dc.description.abstractWe consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit h → 0. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L 2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L 2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which - as it is well known - is not pointwise positive in general.en
dc.description.sponsorshipA. Athanassoulis would like to thank the CMLS, Ecole polytechnique for its hospitality during the preparation of this work. He was also partially supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). F. Pezzotti was partially supported by Project CBDif-Fr ANR-08-BLAN-0333-01.en
dc.publisherEuropean Mathematical Publishing Houseen
dc.subjectHartree dynamicsen
dc.subjectHusimi transformen
dc.subjectSemiclassical analysisen
dc.subjectWigner formalismen
dc.titleStrong semiclassical approximation of Wigner functions for the Hartree dynamicsen
dc.typeArticleen
dc.identifier.journalRendiconti Lincei - Matematica e Applicazionien
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionEcole Polytechnique, Palaiseau, Franceen
dc.contributor.institutionUniversidad del Pais Vasco, Leioa, Spainen
dc.contributor.institutionUniversita degli Studi di Roma La Sapienza, Rome, Italyen
kaust.grant.numberKUK-I1-007-43en
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