Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation

We consider the ground-state properties of an extended one-dimensional Bose gas with pointwise attractive interactions. We take the limit where the interaction strength goes to zero as the system size increases at fixed particle density. In this limit the gas exhibits a quantum phase transition. We compute local correlation functions at zero temperature, both at finite and infinite size. We provide analytic formulas for the experimentally relevant one-point functions $g_2$, $g_3$ and analyze their finite-size corrections. Our results are compared to the mean-field approach based on the Gross-Pitaevskii equation which yields the exact results in the infinite system size limit, but not for finite systems.