Ionization in damped time-harmonic fields

We study the asymptotic behavior of the wavefunction 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-λtcos ωt), where δ is the Dirac distribution. We find the ionization probability in the limit t → ∞ for all λ and ω. The long pulse limit is very singular and for ω = 0, the survival probability is constλ1/3, much larger than O(λ), the one in the abrupt transition counterpart, V(x, t) = δ(x)1{t≥1/λ} where 1 is the Heaviside function.