  Controlled time-decaying harmonic oscillator changes the threshold of decay
order of the potential functions in order to exist the physical wave operators.
This threshold was first reported by Ishida and Kawamoto \cite{IK} for the
non-critical case. In this paper we deal with the critical case. As for the
critical case, the situation changes drastically, and much more rigorous
analysis is required than that of non-critical case. We study this critical
behavior and clarify the threshold by the power of the log growth of the
potential functions. Consequently, this result reveals the asymptotics for
quantum dynamics of the critical case.
