  This paper is concerned with nonlinear Schr\"odinger equations with a
time-decaying harmonic potential. The nonlinearity is gauge-invariant of the
long-range critical order. In [24] and [22], it is proved that the equation
admits a nontrivial solution that behaves like a free solution with a
logarithmic phase correction in the frameworks of both the final state problem
and the initial value problem. Furthermore, a modified scattering operator has
been established in the case without the potential in [15]. In this paper, we
construct a modified scattering operator for our equation by utilizing a
generator of the Galilean transformation. Moreover, we remove a restriction for
the coefficient of the potential which is required in [22].
