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Derivation of Hartree’s theory for generic mean-field Bose systems. (English) Zbl 1316.81095

Summary: 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.

MSC:

81V70 Many-body theory; quantum Hall effect
35Q40 PDEs in connection with quantum mechanics
82B10 Quantum equilibrium statistical mechanics (general)
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