On the derivation of the Hartree equation in the mean field limit: Uniformity in the Planck constant

François Golse (1) , Thierry Paul (1) , Mario Pulvirenti (2)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 UNIROMA - Università degli Studi di Roma "La Sapienza" = Sapienza University [Rome]

In this paper the Hartree equation is derived from the N-body Schrödinger equation uniformly in the Planck constant in two different cases, specifically (a) for Töplitz initial data and Lipschitz interaction force, and (b) for analytic initial data and interaction potential, and over short time intervals, independent of the Planck constant. The convergence rates in these two cases are 1/ log log N and 1/N respectively. The treatment of the second case is entirely self-contained and all the constants appearing in the final estimate are explicit. Moreover it provides a derivation of the Vlasov equation out of the N-body classical dynamics using hierarchies and not empirical measures.

Mots clés: Mecanique quantique; EDP; analyse semiclassique

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

François Golse, Thierry Paul, Mario Pulvirenti. On the derivation of the Hartree equation in the mean field limit: Uniformity in the Planck constant. 2016. ⟨hal-01334365v2⟩