numerical values of the angular momenta in the two cases to be equal, [*i.e.*]{}, we set $J=\ell\,
{\tilde\hbar}$.

In between the ‘kicks’ at a period, and as long as it does not hit the boundary at $r=1$, the electron’s motion is governed by the static Hamiltonian $\tilde H_{dc}$. The equation of motion in this case is $$\ddot r = -\left(\frac{{\tilde\omega_c}}{2}\right)^2 r + 
\frac{J^2}{r^3},$$ whose solution, in terms of the energy $E$, $$\label{eq:t}
E = \frac{1}{2} p_r^2
+ \frac{J^2}{2 r^2} + \frac{1}{2}
\left(\frac{{\tilde\omega_c}}{2}\right)^2 r^2 + J 
\frac{{\tilde\omega_c}}{2},$$ is given by $$\label{eq:u}
\left( \begin{array}{c} 
r(\tau) \\
p_r(\tau) 
\end{array} \right) = 
\left( \begin{array}{c}
\sqrt{\frac{2}{{\tilde\omega_c}}\left[ b + a 
\sin\left\{ {\tilde\omega_c}(\tau-\tau_0) 
+ \sin^{-1}\left(\frac{\frac{1}{2}{\tilde\omega_c} r_0^2 - b}{a}
\right)\right\}\right]} \\
\frac{a}{r(\tau)}\cos\left\{ {\tilde\omega_c}(\tau-\tau_0) + \sin^{-1}
\left(\frac{\frac{1}{2}{\tilde\omega_c} r_0^2 - b}{a}\right)\right\}
\end{array} \right),$$ where $$\label{eq:v}
b = 2E/{\tilde\omega_c} - J \quad {\rm and} \quad a = \sqrt{b^2 - 
J^2}.$$ Here, $r_0$ and $\tau_0$ are initial conditions. For a given energy $E$, the motion is constrained by the centrifugal barrier on one side, and the smaller of the wall radius (equal to 1) and the constraint imposed by the attractive quadratic potential on the other:

\[allw\] $$\label{eq:w1}
r_{min} \le r(\tau) \le {\rm Min}\{r_{max},1\}, \qquad {\rm where}$$ $$\label{eq:w2}
r_{min} = \sqrt{\frac{2}{{\tilde\omega_c}}(b-a)},\qquad {\rm 