Ground state energy of a Bose gas in the Gross-Pitaevskii regime

TL;DR

This paper reviews rigorous mathematical estimates for the ground state energy of dilute Bose gases, focusing on the Gross-Pitaevskii regime, and introduces new bounds combining Dyson and Bogoliubov approaches.

Contribution

It presents a new second-order upper bound for the ground state energy of hard spheres in the Gross-Pitaevskii limit, combining Dyson and Bogoliubov methods.

Findings

Correct leading order asymptotics for hard spheres

Rigorous second-order energy estimate in the Gross-Pitaevskii regime

New upper bound for hard sphere energy in the dilute limit

Abstract

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson's upper bound, which provides the correct leading order asymptotics for hard spheres. Afterwards, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross-Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross-Pitaeavskii limit. Here we only sketch the main ideas, details will appear elsewhere.