the form, $B(t) = B_{dc} + 
B_{ac} f(t)$, where $f(t)=f(t+T_0)$ is some periodically time varying function. We can separate the Hamiltonian $H=H_{dc} + H_1(t)$, where $$\label{eq:b}
H_{dc} = -\frac{{\hbar^2}}{2m^*}\left( \frac{d^2}{d\rho^2} 
+ \frac{1}{\rho}\frac{d}{d\rho}\right) + 
\frac{{\hbar^2\ell ^2}}{2m^*}\frac{1}{\rho^2} + \frac{1}{8} m^* 
\omega_{c}^2(t) \rho^2 + \frac{1}{2} \frac{\ell \hbar \omega_c}{2},$$ and $H_1(t)=\frac{1}{8}m^*\left(2B_{dc}B_{ac}f(t) 
+ B_{ac}^2f^2(t)\right)\rho^2$. Here $H_{dc}$ is the standard static Hamiltonian for a charge in a homogeneous constant perpendicular magnetic field, that includes the para- and dia-magnetic contributions, with $\omega_c = \frac{e B_{dc}}{m^* c}$. With the additional dropping of a term of the form $L_z B_{ac}f(t)$ which can trivially be removed by a unitary transformation, $H_1(t)$ gives the time-dependent contribution to $H$. Note that $H_1(t)=H_1(t+T_0)$. In the limit in which $H_1(t)$ is much smaller than $H_{dc}$ one can study the modification to the solutions associated to $H_{dc}$ by standard time-dependent perturbation theory. As can be seen from the classical phase diagram given in Fig. 1 the boundary between regular and chaotic behavior in fact occurs for $\epsilon=B_{ac}/B_{dc}>1$ and ${\tilde{\omega_c}}>1$. We are then led to approximate $H_1(t)$ by, $$\label{eq:bb}
H_1(t)=\frac{1}{8}m^*\left({(\epsilon\omega_{c})}^2
\sum _{n=-\infty}^{\infty}\delta(t-nT_0)\right)\rho^2.$$ With this simplification, the Hamiltonian (\[eq:a\]) is then approximated by the sum of Eqs. (\[eq:b\]) and (\[eq:bb\]). This choice is also motivated by the 