Low-density limit of dynamical correlations in the Lieb-Liniger model

We derive explicit expressions for dynamical correlations of the field and density operators in the Lieb-Liniger model, within an arbitrary eigenstate with a small particle density ${\cal D}$. They are valid for all space and time and any interaction strength $c>0$, and are the leading order of an expansion in ${\cal D}$. This expansion is obtained by writing the correlation functions as sums over form factors when formally decomposed into partial fractions.