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 cosωt), 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 const λ, 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.