Since we have a periodic system, $\tilde H(\tau+1) = \tilde 
H(\tau)$, from the Floquet theorem[@casati2], it is sufficient to determine the one-period time evolution operator $U(\tau_0 + 1,\tau_0)$, from

\[allo\] $$\label{eq:o1}
i {\tilde\hbar}\,\frac{\partial}{\partial\tau}\,U(\tau) = ( \tilde 
H_{dc} + \tilde V_1(\tau) )\,U(\tau) \, , \quad 0 < \tau \le 1,$$ $$\label{eq:o2}
\tilde V_1(\tau) = \tilde V \, \delta(\tau-1), \qquad {\rm where} 
\qquad 
\tilde V = \frac{1}{2}\ \eta\ r^2,$$

where the parameter $\eta$ has been defined previously. All the information about the dynamics of the system is contained within this Floquet operator, since $\Phi(r,\phi,\tau+1) = U\,
\Phi(r,\phi,\tau)$, where $\Phi$ is the total wave function. Because of the periodic $\delta$-kicked dynamics, we can immediately integrate equation (\[eq:o1\]) to get $$\label{eq:p}
U_\ell(1,0) = \exp\left(-\frac{i}{{\tilde\hbar}}\,\tilde V\right)\,
\exp\left(-\frac{i}{{\tilde\hbar}}\,\tilde H_{dc}\right).$$ The subscript $\ell$ has been attached to $U$ to emphasize that the evolution operator has been restricted to that single $\ell$ value. In other words, states with different values of $\ell$ evolve independently, an immediate consequence of the conservation of angular momentum in this system. The rightmost exponential operator in equation (\[eq:p\]) evolves the wave function from just after the ‘kick’ at $\tau=0$ to just before the kick at one period under the influence of $\tilde H_{dc}$, 