Derivation of Hartree's theory for generic mean-field Bose systems

Mathieu Lewin (1) , Phan Thành Nam (1) , Nicolas Rougerie (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 LPM2C - Laboratoire de physique et modélisation des milieux condensés

In this paper we provide a novel strategy to prove the validity of Hartree's theory for the ground state energy of bosonic quantum systems in the mean-field regime. For the well-known case of trapped Bose gases, this can be shown using the strong quantum de Finetti theorem, which gives the structure of infinite hierarchies of k-particles density matrices. Here we deal with the case where some particles are allowed to escape to infinity, leading to a lack of compactness. Our approach is based on two ingredients: (1) a weak version of the quantum de Finetti theorem, and (2) geometric techniques for many-body systems. Our strategy does not rely on any special property of the interaction between the particles. In particular, our results cover those of Benguria-Lieb and Lieb-Yau for, respectively, bosonic atoms and boson stars.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie. Derivation of Hartree's theory for generic mean-field Bose systems. Advances in Mathematics, 2014, 254, pp.570-621. ⟨10.1016/j.aim.2013.12.010⟩. ⟨hal-00796683v4⟩