Modified scattering operator for nonlinear Schr\"odinger equations with time-decaying harmonic potentials

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].