It is well-known that the Gross-Pitaevskii variational formula describes the ground-state energy per particle of
N-indistinguishable trapped particles (Bosons) in a dilute state in the large system size
N→∞. The goal of the present article is to prove that the Gross--Pitaevskii formula also appears in the iterated limit of zero temperature and large system size of the product ground-state energy per particle of the
N-particle Hamiltonian operator. For fixed particle number
N, we moreover identify the low-temperature limit of the Brownian model with mutual intersection-local-time interaction as the ground-product-state energy per particle of the corresponding
N-body system with repulsive Dirac interaction. This proves the conjectures of Adams, König and Bru from arXiv:math/0411660 and arXiv:math/0512305.
We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number [Formula: see text] becomes large. In the dilute regime, when the interaction potentials have the length scale of order [Formula: see text], 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 [Formula: see text], 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 adaptation to the multi-component setting is non-trivial in various respects and the analysis will be presented in detail.
A fundamental problem in quantum mechanics is to understand the structure and the energy of ground states of interacting systems of many particles. The quantum correlations in ground states or low lying energy states are supposed to explain phenomena such as superfluidity or superconductivity. A long-standing conjecture in mathematical physics has been to establish a universal two-term asymptotic formula for the ground state energy of a system of bosons in the dilute limit of low density predicted by the theory of superfluidity. We discuss a recent proof of this formula.