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Mean field dynamics of Boson stars. (English) Zbl 1113.81032
Summary: We consider a quantum mechanical system of $N$ bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to describe a condensate where the $N$ bosons are all in the same one-particle state. Starting from the $N$-body Schrödinger equation, we prove that, in the limit $N\to \infty$, the time evolution of the one-particle density is governed by the relativistic nonlinear Hartree equation. This equation is used to describe the dynamics of boson stars (Chandrasekhar theory). The corresponding static problem was rigorously solved in [E. Lieb and H.-T. Yau, Commun. Math. Phys. 112, No. 1, 147–174 (1987)].

##### MSC:
 81Q05 Closed and approximate solutions to quantum-mechanical equations 85A15 Galactic and stellar structure