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We continue our study of the emergence of Non-Equilibrium Steady States inquantum integrable models focusing on the expansion of a Lieb-Liniger gas forarbitrary repulsive interaction. As a first step towards the derivation of theasymptotics of observables in the thermodynamic and large distance and timelimit, we derive an exact multiple integral representation of the time evolvedmany-body wave-function. Starting from the known but complicated expression forthe overlaps of the initial state of a geometric quench, which are derived fromthe Slavnov formula for scalar products of Bethe states, we eliminate theawkward dependence on the system size and distinguish the Bethe states intoconvenient sectors. These steps allow us to express the rather impractical sumover Bethe states as a multiple rapidity integral in various alternative forms.Moreover, we examine the singularities of the obtained integrand and calculatethe contribution of the multivariable kinematical poles, which is essentialinformation for the derivation of the asymptotics of interest.

Non-Equilibrium Steady State of the Lieb-Liniger model: multiple-integral representation of the time evolved many-body wave-function