We first study the asymptotic behavior of the wave function in a 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→∞ for all λ and ω. The long pulse limit is very singular, and, for ω = 0, the survival probability is constant λ1/3 (λ is a small parameter), much larger than O(λ), the one in the abrupt transition counterpart.
Later we study the asymptotic behavior of the wave function in a one-dimensional model on half line of ionization by pulses, where the time-dependent potential is of the form V(x, t) = δ(x-1)(V + 2Ω sin(ωt)).
For Ω=0, we give a complete asymptotic decomposition to wave function in the limit t→∞ for all V and ω; for a nonzero Ω we give a partial decomposition, from which we give an equation that determines the condition when ionization happens.