radius $R_0$, all masses by the effective mass $m^*$ and all times by the period of the $a.c.$ field, $T_0$. Thus, we define

\[allf\] $$\label{eq:f1}
r = \rho/R_0; \qquad 0\le r \le 1,$$ $$\label{eq:f2}
\tau = t/T_0 \equiv \frac{\omega_0}{2\pi}t, \qquad
{\tilde\omega_c} = \omega_c/\omega_0, \qquad
{\tilde\hbar} = \frac{\hbar}{m^*\omega_0 R_0^2}.$$

In these units, equations (\[eq:b\]) and (\[eq:bb\]) become

\[allg\] $$\label{eq:g0}
\tilde H = \tilde H_{dc} + \tilde H_1(\tau)$$ $$\label{eq:g1}
\tilde H_{dc} = -\frac{{\tilde\hbar}^2}{2}\left( \frac{d^2}{dr^2} 
+ \frac{1}{r}\frac{d}{dr}
\right) + \frac{\ell^2 {\tilde\hbar}^2}{2 r^2} + \frac{1}{2}
\left(\frac{{\tilde\omega_c}}{2}\right)^2 r^2 + \ell\, {\tilde\hbar} 
\frac{{\tilde\omega_c}}{2} ,$$ $$\label{eq:g2}
\tilde H_1(\tau) = \frac{1}{2}\ \eta\ r^2 
\sum_{n=-\infty}^{\infty} \delta (\tau-n), \quad {\rm where}$$ $$\eta = \left(\frac{\epsilon\,{\tilde\omega_c}}{2}\right)^2,$$

and the corresponding solutions to the time-independent part, along with the boundary and normalization conditions, are given by

\[allh\] $$\label{eq:h1}
\tilde H_{dc}\, \tilde\Psi_{n\ell}(r,\phi) = \tilde E_{n\ell}\, 
\tilde\psi_{n\ell}(r)  \frac{e^{i \ell \phi}}{\sqrt{2 \pi}},$$ $$\label{eq:h2}
\tilde\Psi(r,\phi) = \sum_{n=1}^{\infty} \sum_{\ell=-\infty}^{\infty} 
\tilde\psi_{n\ell}(r) \frac{e^{i \ell \phi}}{\sqrt{2 \pi}},$$ $$\label{eq:h3}
\tilde\psi_{n\ell}(r = 1) = 0, \qquad {\rm and} \qquad 
\int_{0}^{1} \tilde\psi_{n\ell}^{2}(r) \, r \,dr = 1.$$

As was first pointed out by Dingle[@dingle], the physically acceptable solutions to equations (\[allh\]) are the Whittaker functions of the first kind[@abramowitz], $$\label{eq:i}
\tilde\psi_{n\ell}(r) = \sqrt{\frac{2}{N_{n\ell}}}\; 
\frac{1}{r}\;M_{\chi_{n\ell},{\mid \ell\mid}/2}(2\pi F\, r^2),$$ where the frustration parameter $F = \frac{\Phi}{\Phi_{0}}$ is 