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

Asymptotic behavior for nonlinear Schr\"{o}dinger equations with critical time-decaying harmonic potential