ar X iv : m at h - ph / 0 60 90 69 v 1 2 5 Se p 20 06 Ionization in a 1 - Dimensional Dipole Model

We study the evolution of a one dimensional model atom with δ-function binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ω-periodic real-valued function. Starting with an initially localized state and E(t) a trigonometric polynomial, complete ionization occurs (the probability of finding the electron in any fixed region goes to zero). For more general periodic fields and ψ(x, 0) compactly supported (this is a technical point making the exposition cleaner), we construct a resonance expansion. More precisely, we prove that ψ(x, t) has a unique decomposition into a quasi-bound state e −iσ b t ψ b (x, t) and a dispersive component ψ d (x, t) (both square integrable in space, with σ b and ψ b (x, t) independent of ψ(x, 0)). The quasi-bound state ψ b (x, t) is 2π/ω periodic in time and exponentially decaying in space. The dispersive part is given by a Borel summable asymptotic power series in t −1/2 with coefficients varying with x. In the event ℑσ b = 0, then ψ b (x, t) is a Floquet eigenstate and orthogonal to ψ d (x, t).