Ground state energy of mixture of Bose gases

TL;DR

This paper analyzes the ground state energy of multi-component Bose gases in different regimes, confirming the validity of the Gross-Pitaevskii functional and Bogoliubov's approximation, and extending these results to complex multi-component systems.

Contribution

It extends the analysis of ground state energies to multi-component Bose gases, verifying the Gross-Pitaevskii and Bogoliubov approximations in different regimes, which was previously established mainly for single-component systems.

Findings

Gross-Pitaevskii energy functional accurately captures leading order energy in dilute regime

Many-body ground state fully condenses on Gross-Pitaevskii minimizers

Bogoliubov's approximation verified with second order energy expansion in mean-field regime

Abstract

We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number $N$ becomes large. In the dilute regime, when the interaction potentials have the length scale of order $O(N^{-1})$, we show that the leading order of the ground state energy is captured correctly by the Gross-Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross-Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is $O(1)$, we are able to verify Bogoliubov's approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaption to the multi-component setting is non-trivial in various respects and the analysis will be presented in details.