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Higher order corrections to the mean-field description of the dynamics of interacting bosons. (English) Zbl 1439.82029

Summary: In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of \(N\) \(d\)-dimensional bosons for large \(N\). The bosons initially form a Bose-Einstein condensate and interact with each other via a pair potential of the form \((N-1)^{-1} N^{d \beta}v(N^\beta \cdot)\) for \(\beta \in [0,\frac{1}{4d})\). We derive a sequence of \(N\)-body functions which approximate the true many-body dynamics in \(L^2({\mathbb{R}}^{dN})\)-norm to arbitrary precision in powers of \(N^{-1}\). The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
81V10 Electromagnetic interaction; quantum electrodynamics
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
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