Asymptotic behavior for nonlinear Schrödinger equations with critical time-decaying harmonic potential

Time-decaying harmonic oscillators yield dispersive estimates with weak decay, and change the threshold power of nonlinearity between short long range. In non-critical case for time-decaying oscillator, this can be characterized by polynomial nonlinearities. However, in critical case, it is difficult to characterize using only terms, thus we use logarithmic nonlinear terms.