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 - cited By 4

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.

U2 - 10.4171/RLM/613

DO - 10.4171/RLM/613

M3 - Article

VL - 22

SP - 525

EP - 552

JO - Rendiconti Lincei Matematica e Applicazioni

JF - Rendiconti Lincei Matematica e Applicazioni

SN - 1120-6330

IS - 4

ER -