Ground state energy of the $δ$-Bose gas at weak coupling from double extrapolation

Sylvain Prolhac

Published 2016-10-12Version 1

We consider the ground state energy $e(\gamma)$ of the $\delta$-Bose gas with repulsive interaction in the weak coupling regime $\gamma\to0$, where previous analytical studies gave the expansion $e(\gamma)\simeq\gamma-4\gamma^{3/2}/3\pi+(1/6-1/\pi^{2})\gamma^{2}$. Using a numerical solution of the Bethe equations for a finite number $N$ of particles and finite strength $\gamma$ of the interaction, we obtain very accurate numerics for the next orders after extrapolating on $N$ and $\gamma$. The coefficient of $\gamma^{5/2}$ in the expansion is found approximately equal to $-0.00158769986550594499$. This value is supported by a numerical solution of the Lieb-Liniger integral equation discretized with $M$ points followed by extrapolation on $M$ and $\gamma$.