Ionization in damped time-harmonic fields

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 $t\to \mathrm{\infty }$ for all $\lambda$ and $\omega$. The long pulse limit is very singular, and, for $\omega =0$, the survival probability is $const \lambda^{1/3}$,muchlargerthan$O(\lambda)$, the one in the abrupt transition counterpart, $V(x,t)=\delta (x){\mathbf{1}}_{\{t\ge 1\mathrm{/}\lambda \}}$ where $\mathbf{1}$ is the Heaviside function.