Strong semiclassical approximation of Wigner functions for the Hartree dynamics

Type
Article

Authors
Athanassoulis, Agissilaos
Paul, Thierry
Pezzotti, Federica
Pulvirenti, Mario

KAUST Grant Number
KUK-I1-007-43

Date
2011

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.

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.

Publisher
European Mathematical Society - EMS - Publishing House GmbH

Journal
Rendiconti Lincei - Matematica e Applicazioni

DOI
10.4171/RLM/613

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