Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime

TL;DR

This paper proves that a trapped Bose gas exhibits complete Bose-Einstein condensation at an optimal rate, extending previous results to non-translation invariant systems without small scattering length assumptions.

Contribution

It generalizes prior work by establishing optimal condensation rates for trapped bosons in the Gross-Pitaevskii regime, removing previous restrictions.

Findings

Proves complete Bose-Einstein condensation with optimal rate.

Extends results to trapped systems beyond translation invariance.

Removes smallness assumption on scattering length.

Abstract

We consider a Bose gas consisting of $N$ particles in $\mathbb{R}^3$, trapped by an external field and interacting through a two-body potential with scattering length of order $N^{-1}$. We prove that low energy states exhibit complete Bose-Einstein condensation with optimal rate, generalizing previous work in \cite{BBCS1, BBCS4}, restricted to translation invariant systems. This extends recent results in \cite{NNRT}, removing the smallness assumption on the size of the scattering length.