TY - JOUR

T1 - Strong semiclassical approximation of Wigner functions for the Hartree dynamics

AU - Athanassoulis, Agissilaos

AU - Paul, Thierry

AU - Pezzotti, Federica

AU - Pulvirenti, Mario

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/599749

UR - http://www.ems-ph.org/doi/10.4171/RLM/613

UR - http://www.scopus.com/inward/record.url?scp=84858657726&partnerID=8YFLogxK

U2 - 10.4171/RLM/613

DO - 10.4171/RLM/613

M3 - Article

VL - 22

SP - 525

EP - 552

JO - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni

JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni

SN - 1120-6330

IS - 4

ER -