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Derivation of the Hartree equation for compound Bose gases in the mean field limit

Ioannis Anapolitanos

Institute for Analysis, Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany

E-mail Address: ioannis.anapolitanos@kit.edu

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Michael Hott

Department of Mathematics, University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA

E-mail Address: mhott@math.utexas.edu

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Dirk Hundertmark

Institute for Analysis, Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany

E-mail Address: dirk.hundertmark@kit.edu

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https://doi.org/10.1142/S0129055X17500222Cited by:20 (Source: Crossref)

Abstract

We consider mixtures of Bose gases of different species. We prove that in the mean field limit and under suitable conditions on the initial condition, a system composed of two Bose species can be effectively described by a system of coupled Hartree equations. Moreover, we derive quantitative bounds on the rates of convergence of the reduced density matrices in Sobolev trace norms. We treat both the non-relativistic case in the presence of an external magnetic field $A \in L_{loc}^{2} (ℝ^{3}; ℝ^{3})$ and the semi-relativistic case.

Keywords:

AMSC: 35Q40, 35Q41, 81V70

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