  The Lieb-Liniger model which has a weak external potential term under the
periodic boundary condition is investigated. By exploiting the Bethe states as
bases, we perform a perturbation analysis up to the first order to obtain the
shifts of eigenenergies and corresponding eigenstates which have been brought
about by the external potential. If we take a sufficiently large system, the
eigenstates can be "the Schr\"odinger's cat states". Expectation values of the
density operator taken between two Bethe states can be calculated with the aid
of the Slavnov's formula and we evaluate the influence of the many-body
interaction to the system under the external potential. The system is
insensitive to the external potential because of the many-body interaction.
