Correlation functions of the Lieb-Liniger gas and the LeClair-Mussardo formula

In this letter we derive formulas for multi point correlation functions, in the thermodynamic limit, for the Lieb Liniger gas taken with respect to arbitrary eigenstates. These results apply for the ground state, thermal states and GGE states. We obtain these correlation functions as a series of multiple integrals of progressively higher dimensions. These integrals converge rapidly for short distance correlation functions and low densities of particles. The series derived matches exactly the LeClair Mussardo formula for correlation functions of relativistic integrable models.