  We study the asymptotic behavior of the wave function in a simple one
dimensional model of ionization by pulses, in which the time-dependent
potential is of the form $V(x,t)=-2\delta(x)(1-e^{-\lambda t} \cos\omega t)$,
where $\delta$ is the Dirac distribution. We find the ionization probability in
the limit $t\to\infty$ for all $\lambda$ and $\omega$. The long pulse limit is
very singular, and, for $\omega=0$, the survival probability is $const
\lambda^{1/3}$, much larger than $O(\lambda)$, the one in the abrupt transition
counterpart, $V(x,t)=\delta(x)\mathbf{1}_{\{t\ge 1/\lambda\}}$ where
$\mathbf{1}$ is the Heaviside function.
