$x_1<0$ and $x_2>0$.

![(Color online) Plots of the potential $\Omega_N (\alpha; R, \rho)$ \[Eq. (\[PotentialDeff\])\] for different $\alpha$ and $N$. Roman numerals in plot (a) label quadrants. Black solid lines denote the level sets of the expectation value of the initial energy, $\bar{E}_N$ \[Eq. (\[AvEDef\])\], i.e., the boundary between the classically allowed and classically forbidden regions. (a) $N=1$ and $\alpha=3$; (b) $N=1$ and $\alpha = -3$; (c) $N=2$ and $\alpha=3$; (d) $N=2$ and $\alpha=-3$; (e) $N=4$ and $\alpha=3$; (f) $N=4$ and $\alpha=-3$. []{data-label="Fig_potentials"}](Potentials.pdf)

Before stating the results of numerical calculations, let us qualitatively analyze dynamics of the system within a semiclassical consideration. Figure \[Fig\_potentials\] presents the plots of the potentials (\[PotentialDeff\]). According to the initial condition \[see Eqs. (\[InitialCondition\]) and (\[AvEDef\])\], the particle is located on the axis $\rho=0$ and its initial velocity is directed along this axis toward the first quadrant, and the amplitude of the velocity is chosen such that the total energy of the particle equals $\bar{E}_N$; hence, the boundaries between the classically allowed and classically forbidden regions are drawn by solid black lines in Fig. \[Fig\_potentials\]. Now compare Fig. \[Fig\_potentials\](a) with Fig. \[Fig\_potentials\](b). Since the semi classical counterpart of our quantum particle “experiences” the barrier in 