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 $V(x,t)=-2δ(x)(1-e^{-λt} \cosωt)$, where $δ$ is the Dirac distribution. We find the ionization probability in the limit $t\to\infty$ for all $λ$ and $ω$. The long pulse limit is very singular, and, for $ω=0$, the survival probability is $const λ^{1/3}$, much larger than $O(λ)$, the one in the abrupt transition counterpart, $V(x,t)=δ(x)\mathbf{1}_{\{t\ge 1/λ\}}$ where $\mathbf{1}$ is the Heaviside function.