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

作者：Lewin, Mathieu^*; Phan Thanh Nam; Rougerie, Nicolas

来源：Advances in Mathematics, 2014, 254: 570-621.

DOI：10.1016/j.aim.2013.12.010

• Hartree theory
• Many-particle Hamiltonian
• Mean-field regime
• Quantum de Finetti theorem
• Bosonic atoms
• Boson stars

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 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.

• 出版日期2014-3-20