the ratio of the flux threading the dot to the quantum of flux $\Phi_0=h/2e$. The quantities $\chi_{n\ell}$ are related to the eigenvalues via $$\label{eq:k}
\chi_{n\ell} = \frac{1}{2}( \tilde E_{n\ell} - \ell ),$$ and are determined precisely by the requirement that the wavefunction vanishes at the boundary, equation (\[eq:h3\]), $M_{\chi_{n\ell},{\mid \ell\mid}/2}(F) = 0$. In the limit of no confinement, $R_0 \rightarrow \infty$, we recover the usual Laguerre polynomial solutions for the $\tilde\psi_{n\ell}$’s.

The frustration parameter $F$ can also be written as $$\label{eq:m}
F = \frac{1}{4\pi}\left(\frac{R_0}{\ell_H}\right)^2, \qquad 
{\rm where} 
\qquad \ell_H=\left(\frac{\hbar c}{eB_{dc}}\right)^{1/2},$$ that is, it’s proportional to the square of the ratio of the confinement radius to the magnetic length. When $2\pi F \ll 1$, the problem is equivalent to that of a nearly free electron, bound by a very weak magnetic field, and so is amenable to a perturbative treatment. In the opposite limit, the boundary can essentially be neglected, and we recover the results of Dingle mentioned previously. It is in the intermediate regime, when the two lengths are comparable, that we expect the effects of confinement to be nontrivial, especially in the presence of strong time-dependent fields.

In principle, we are able to cover the entire range of parameter 