Strong semiclassical approximation of Wigner functions for the Hartree dynamics

Abstract

We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit → 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.

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

“…This convergence rate is established under the assumption that the interaction force field ∇V is bounded and Lipschitz continuous, and involves the Lipschitz constant of ∇V . Observe that the large-time growth of the convergence rate obtained in Theorem 2.5 is exponential, at variance with the estimates obtained in [1,5], which are super-exponential. Our result can be also formulated as a direct comparison between the Vlasov solution and the Husimi function of the Hartree solution.…”

“…Besides, convergence is proved along sequences n → 0. Under additional assumptions on the initial data and the interaction potential, one can prove [1] that the Wigner function of the solution of Hartree's equation is L 2 -close to its weak limit, i.e. to the solution of the Vlasov equation.…”

Semiclassical Limit to the Vlasov Equation with Inverse Power Law Potentials

“…The issue of exhibiting explicit bounds has been addressed in [3,36,1,2,17,19,18]. More precisely, bounds on the rate of convergence of the Hartree evolution towards the Vlasov equation have been first obtained in [3], where the convergence is established in Hilbert-Schmidt norm with a relative rate ε 2/7 for regular initial data and smooth potentials V . Moreover, for smooth interactions, it has been proven in [36,1,2] that the solution to the Hartree equation can be written as an expansion -with no control on the remainder -in powers of ε.…”

Mean-Field Evolution of Fermions with Singular Interaction

“…(1. 2) In (1.2) the choice ε = N −1/3 ensures the kinetic and the potential energy associated to (1.1) to be of comparable order, namely O(N ). Observe that, in contrast with the bosonic case, the mean-field scaling for fermions comes coupled with a semiclassical limit (notice that here ε plays the role of ).…”