  We continue our study of the emergence of Non-Equilibrium Steady States in
quantum integrable models focusing on the expansion of a Lieb-Liniger gas for
arbitrary repulsive interaction. As a first step towards the derivation of the
asymptotics of observables in the thermodynamic and large distance and time
limit, we derive an exact multiple integral representation of the time evolved
many-body wave-function. Starting from the known but complicated expression for
the overlaps of the initial state of a geometric quench, which are derived from
the Slavnov formula for scalar products of Bethe states, we eliminate the
awkward dependence on the system size and distinguish the Bethe states into
convenient sectors. These steps allow us to express the rather impractical sum
over Bethe states as a multiple rapidity integral in various alternative forms.
Moreover, we examine the singularities of the obtained integrand and calculate
the contribution of the multivariable kinematical poles, which is essential
information for the derivation of the asymptotics of interest.
