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Article Dans Une Revue Rendiconti Lincei. Matematica e Applicazioni Année : 2011

Strong semiclassical approximation of Wigner functions for the Hartree dynamics.

Résumé

We consider the Wigner equation corresponding to a nonlinear Schrodinger 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.

Dates et versions

hal-00851542 , version 1 (14-08-2013)

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Citer

Thierry Paul, Agissilaos Athanassoulis, Federica Pezzotti, Mario Pulvirenti. Strong semiclassical approximation of Wigner functions for the Hartree dynamics.. Rendiconti Lincei. Matematica e Applicazioni, 2011, 22 (4), pp.525-552. ⟨10.4171/RLM/613⟩. ⟨hal-00851542⟩
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