  Following the paper by M. Combescure [Ann. Phys. (NY) 204, 113 (1990)], we
apply the quantum singular time dependent oscillator model to describe the
relative one dimensional motion of two ions in a trap. We argue that the model
can be justified for low energy excited states with the quantum numbers $n\ll
n_{max}\sim 100$, provided that the dimensionless constant characterizing the
strength of the repulsive potential is large enough, $g_*\sim 10^5$. Time
dependent Gaussian-like wave packets generalizing odd coherent states of the
harmonic oscillator, and excitation number eigenstates are constructed. We show
that the relative motion of the ions, in contradistinction to its center of
mass counterpart, is extremely sensitive to the time dependence of the binding
harmonic potential, since the large value of $g_*$ results in a significant
amplification of the transition probabilities between energy eigenstate even
for slow time variations of the frequency.
