  By controlling coefficients and decaying order of time-decaying harmonic
potentials, the velocity of a quantum particle is decelerated by the effect of
harmonic potentials but the particle is non-trapping. In this paper, we
consider the quantum system with controlled harmonic potentials. By defining
the wave operators for this system and suitable range of these, we can prove
the existence and completeness of wave operators with respect to the
short-range potentials $V(t,x)$ satisfying $|V(t,x)| = o ((1+|x|)^{-1/(1-
\lambda)})$, for some $\lambda$, $0< \lambda < 1/2$.
