Asymptotic Expansions for Weakly Interacting Bosons

I will give a summary of recent results on asymptotic expansions for non-relativistic bosons in the mean-field limit. These expansions are around Bogoliubov theory and provide approximations for the Bose gas to any order in the small parameter $N^{-1/2}$, where $N$ is the number of particles. The expansions were proven for both the dynamics, and the low-lying energies and eigenstates for particles with pair interaction. We have proven an additional result for the dynamics of bosons that are coupled linearly to a Nelson-type quantum field. The talk will in particular emphasize the explicit nature of the expansion, i.e., how the expansion can be used to compute physical quantities such as the binding energy, in a way that is independent of the particle number $N$. We also explain the connection to Edgeworth expansions in probability theory.

This is joint work with Lea Bossmann, Marco Falconi, Nikolai Leopold, David Mitrouskas, Peter Pickl, Robert Seiringer, and Avy Soffer.