# Complete Bose-Einstein condensation in the Gross-Pitaevskii regime

## TL;DR

This paper proves that in the Gross-Pitaevskii regime, a bosonic gas exhibits complete Bose-Einstein condensation in the ground state and low-energy states, with a uniform bound on excitations, assuming a small interaction potential.

## Contribution

It establishes rigorous proof of complete Bose-Einstein condensation in the Gross-Pitaevskii limit under small potential assumptions.

## Key findings

- Ground state shows complete Bose-Einstein condensation.
- All low-energy states have a bounded number of excitations.
- Condensation is uniform across the system, independent of particle number.

## Abstract

We consider a gas of $N$ bosons in a box with volume one interacting through a two-body potential with scattering length of order $N^{-1}$ (Gross-Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently small, we show that the ground state of the system and all states with relatively small excitation energy exhibit complete Bose-Einstein condensation, with a uniform (i.e. $N$ independent) bound on the number of excitations.