while the operator to it’s left propagates it from just before to just after the kick at a period.

Since $U$ is an Unitary operator, the spectrum of it’s eigenvalues can be represented as $$\label{eq:q}
U_\ell\,\phi_{n\ell} = e^{i\,\varepsilon_{n\ell}}\,\phi_{n\ell}.$$ The set of eigenvalues $\{\varepsilon_{n\ell} \in (0,2\pi]\}$, are collectively known as the Quasi-energy Eigenvalues (QEE), and the eigenfunctions $\{\phi_{n\ell}\}$ as the Quasi-energy Eigenfunctions (QEF) of $U$. The investigation of the quantum dynamics of the system is completely equivalent to determining the nature of the QEE and QEF. The fundamental task is thus to obtain the Quasi-energy Spectrum (QES) of the evolution operator given by equation (\[eq:p\]).

Classical Dynamics
==================

We begin the discussion of the behavior of the model by looking at it’s classical dynamics. The classical Hamiltonian corresponding to the quantum one given by equations (\[allg\]) is,

\[alls\] $$\label{eq:s0}
\tilde H = \tilde H_{dc} + \tilde H_1(\tau)$$ $$\label{eq:s1}
\tilde H_{dc} = \frac{1}{2} p_r^2
+ \frac{J^2}{2 r^2} + \frac{1}{2}
\left(\frac{{\tilde\omega_c}}{2}\right)^2 r^2 + J 
\frac{{\tilde\omega_c}}{2} ,$$ $$\label{eq:s2}
\tilde H_1(\tau) = \frac{1}{2}\ \eta\ r^2 
\sum_{n=-\infty}^{\infty} \delta (\tau-n),$$

where $p_r$ is the radial momentum and $J$ is the [*conserved*]{} angular momentum. To make quantitative correspondences between the classical and quantum results, we always set the 