  We analyze the long time behavior of solutions of the Schr\"odinger equation
$i\psi_t=(-\Delta-b/r+V(t,x))\psi$, $x\in\RR^3$, $r=|x|$, describing a Coulomb
system subjected to a spatially compactly supported time periodic potential
$V(t,x)=V(t+2\pi/\omega,x)$ with zero time average. We show that, for any
$V(t,x)$ of the form $2\Omega(r)\sin (\omega t-\theta)$, with $\Omega(r)$
nonzero on its support, Floquet bound states do not exist. This implies that
the system ionizes, {\em i.e.} $P(t,K)=\int_K|\psi(t,x)|^2dx\to 0$ as
$t\to\infty$ for any compact set $K\subset\RR^3$. Furthermore, if the initial
state is compactly supported and has only finitely many spherical harmonic
modes, then $P(t,K)$ decays like $t^{-5/3}$ as $t \to \infty $. To prove these
statements, we develop a rigorous WKB theory for infinite systems of ordinary
differential equations.
