On the mean-field approximation of many-boson dynamics

Abstract

We show under general assumptions that the mean-field approximation for quantum many-boson systems is accurate. Our contribution unifies and improves most of the known results. The proof uses general properties of quantization in infinite dimensional spaces, phase-space analysis and measure transportation techniques.

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

“…where W : R d → R is an even measurable function and V is a real-valued potential both satisfying the following assumptions for some p and q, Remark that the assumption on W are satisfied for instance by the Coulomb type potentials λ |x| α when 0 < α < 2, λ ∈ R and d = 3. For more details we refer the reader to [39], where Thm. 2.3 and 2.4 are applied to the mean-field theory of quantum many-body dynamics.…”

“…So, we deduce that v(t, γ(t)) ∈ L 1 (I, Z 0 ) for η-a.e. 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.…”

“…||γ i || Z0 ) .Notice that the case M = 0 is trivial. The hypothesis(15) with the Duhamel formula(39), give the existence of a constant C(M,Ī) > 0 such that for any t ∈Ī,||γ 1 (t) − γ 2 (t)|| Z0 ≤ t s ||v(τ, γ 1 (τ )) − v(τ, γ 2 (τ ))|| Z0 dτ ≤ C(M,Ī) t s (||γ 1 (τ )|| r Z1 + ||γ 2 (τ )|| r Z1 ) ||γ 1 (τ ) − γ 2 (τ )|| Z0 dτ.…”

On the general principle of the mean-field approximation for many-boson dynamics

“…for any (φ 1 , ..., φ p , ψ 1 , ..., ψ p ) ∈ Q(A) 2p . Then as shown in [54,Lemma 3.1], one can extend each q (p) i,j to a unique symmetric quadratic form on Q(H 0 N ). Each form q (p) i,j represents the two-body interaction between the i-th and j-th particles.…”

“…The method of Wigner measures was used for the study of Schrödinger many-boson dynamics with singular potentials of Coulomb type in [12]. Subsequently, the latter result was improved in [54] where a quite general framework is presented. Here, we build upon the work of Q. Liard and prove that the mean field approximation actually relies only in some elementary principles.…”