Mean field limit for bosons with compact kernels interactions by Wigner measures transportation

Abstract

We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in [4,6,11].

Concentration of cylindrical Wigner measures

“…In addition, the projective family of measures indexed by spaces of even codimension is a projective subfamily of (µ Ψ ) Ψ∈F (X ′ w ) with cofinal index set. By a theorem of Prokhorov [21,Theorem I.21], (20) is then sufficient to prove that µ is concentrated as a Radon measure on X ′ w . ⊣…”

On the mean-field approximation of many-boson dynamics

“…where V is a real measurable function, W is a real potential, satisfying W (−x) = W (x), x ∈ R d . It is in principle meaningful to include multi-particles interactions but to keep the presentation simple we avoid to do so (see [7,35,16]). Since we are dealing with bosons we assume that H N is a self-adjoint operator on the symmetric tensor product space L 2 s (R dN ).…”

On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs

“…Then the Duhamel formula (39) implies that η concentrates actually on absolutely continuous curves γ ∈ W 1,1 (I, Z 0 ). Furthermore, using the estimate (40), with p = r, we see also that γ ∈ L r (I, Z 1 ) for η-a.e. Therefore, the measure η concentrates on the solutions γ ∈ L r (I, Z 1 ) ∩ W 1,1 (I, Z 0 ) of the initial value problem (10).…”

“…If we consider the classical limit, → 0 where is an effective "Planck constant" which depends on the scaling of the system at hand, then quantum states transform in the limit → 0 into probability measures satisfying a Liouville equation related to a nonlinear Hamiltonian PDE, see [9,10,11,12]. Therefore, the uniqueness property for probability measure solutions of Liouville's equation is a crucial step towards a rigourous justification of the classical limit or the so-called Bohr's correspondence principle for quantum field theories [7,8,12,38,40].…”