  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.
