Commutative diagram of the Gross-Pitaevskii approximation

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 $N\to \mathrm{\infty }$. 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, K"onig and Bru from arXiv:math/0411660 and arXiv:math/0512305.