Seminar Analysis and Mathematical Physics: Soeren Petrat (Jacobs University Bremen)

All Orders of the Dynamics of Bosons in the Mean-field Limit

We consider the non-relativistic quantum dynamics of N bosons in the mean-field scaling limit. It is known that the leading order behavior is described by the Hartree equation, and the next-to-leading order by Bogoliubov theory. Here, we prove a perturbative expansion around Bogoliubov theory: a norm approximation of the true solution to the Schroedinger equation to any order in 1/N. The coefficients in the expansion are independent of N, and can be computed from the solutions to the Hartree and Bogoliubov equations alone. Our expansion leads to approximations of correlation functions and reduced densities to any order in 1/N. In this sense we have completely solved the dynamics of this mean-field model, at least for bounded interaction potentials.

This is joint work with Lea Bossmann, Peter Pickl, and Avy Soffer.

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