(where $x_1 < 0$ and $x_2 < 0$) to the first quadrant (where $x_1 > 0$ and $x_2 > 0$); see Fig. \[Fig\_potentials\]. While the numerical calculations are done for a specific Hamiltonian, the analytical analysis that follows relies exclusively on the symmetry properties of the 2D potential, making our conclusions, drawn from the numerical analysis, general.

The model Hamiltonian for our system is chosen as (atomic units are used throughout) $$\begin{aligned}
&& \hat{H}_N (\alpha) = -\frac 1{2} \left( \frac{\partial^2}{\partial x_1^2} +   \frac{\partial^2}{\partial x_2^2} \right) + \Omega_N (\alpha; x_1, x_2), \label{HamiltonianCartesianDef} \\
&& \Omega_N (\alpha; x_1, x_2) = \alpha V(x_1) + 3V(x_2) + U_N(x_2 - x_1). \end{aligned}$$ where $N=1,2,4$ and $\alpha$ being an arbitrary real parameter. The potentials $V(x_1)$ and $V(x_2)$ describe the potential barriers near the origin, for the motion along each of the two coordinates. The parameter $\alpha$ allows us to vary the relative height of the barriers. We have chosen $$\begin{aligned}
V(x) = x \exp(-x^2),\end{aligned}$$ which corresponds to a potential barrier preceded by a potential well.

The potential $U_N(\rho)$ describes the coupling between the two degrees of freedom. In the absence of $U_N(\rho)$ \[i.e., for $U_N(\rho)=0$\], the 2D dynamics breaks into two uncoupled 1D motions. Nontrivial features 