Commutative diagram of the Gross-Pitaevskii approximation

TL;DR

This paper proves that the Gross-Pitaevskii formula, which describes the ground state energy of dilute bosonic systems, also emerges in the combined limit of zero temperature and large particle number through an iterative approach.

Contribution

It establishes a new connection between the Gross-Pitaevskii approximation and the zero-temperature, large-system limit of the product ground state energy for bosons.

Findings

Gross-Pitaevskii formula appears in the iterative zero-temperature and large-system limit.

The result links variational energy formulas with thermodynamic limits.

Provides a new proof technique for the emergence of the Gross-Pitaevskii approximation.

Abstract

It is well-known that the {\it Gross-Pitaevskii} variational formula describes the the ground state energy of of $N$-indistinguishable trapped particles (bosons) in a dilute state in the large system size $N\to\infty$. The goal of the present article is to prove that the Gross-Pitaevskii formula also appears in the {\it iterative limit} of zero temperature and large system size of the {\it product ground state energy} of the $N$-particle Hamiltonian operator.