Nonperturbative Time Dependent Solution of a Simple Ionization Model

Abstract

A. We present a non-perturbative solution of the Schrödinger equation iψt(t, x) = −ψxx(t, x) − 2(1 + α sin ωt)δ(x)ψ(t, x), written in units in which = 2m = 1, describing the ionization of a model atom by a parametric oscillating potential. This model has been studied extensively by many authors, including us. It has surprisingly many features in common with those observed in the ionization of real atoms and emission by solids, subjected to microwave or laser radiation. Here we use new mathematical methods to go beyond previous investigations and to provide a complete and rigorous analysis of this system. We obtain the Borel-resummed transseries (multi-instanton expansion) valid for all values of α, ω, t for the wave function, ionization probability, and energy distribution of the emitted electrons, the latter not studied previously for this model. We show that for large t and small α the energy distribution has sharp peaks at energies which are multiples of ω, corresponding to photon capture. We obtain small α expansions that converge for all t, unlike those of standard perturbation theory. We expect that our analysis will serve as a basis for treating more realistic systems revealing a form of universality in di erent emission processes.Physically, |θ| 2 gives the probability of nding the particle in the eigenstate u b (x) and |Θ(k, t)| 2 is the probability density of the ionized electron in "quasi-free" states (continuous spectrum) with energies k 2 .

Ionization by an Oscillating Field: Resonances and Photons

“…It was also shown in [6] that lim t→∞ Θ(k, t) := Θ(k, ∞) exists. For small α and ω > 1 it has the Lorentzian shape: From (8) it follows that, for α → 0, after t → ∞, Θ becomes a delta function at k 2 = ω − 1.…”

“…(where the square root is chosen so that √ u equals −i |u| if u < 0). Methods similar to those of [6] show that equation (14) has a unique square-summable solution g n = g n (α, σ), analytic in α for small α and real analytic for all α. It is also analytic in σ, except for a square root branch points at 0 and for a pole of order one in the lower half plane; its residue can be calculated using a convergent continued fraction.…”

“…These show, for the rst time we believe, resonances in the energy distribution |Θ(k, t)| 2 which correspond to multiphoton absorption for very long times. These are based on exact expressions in the form of multi-instanton expansions [6]. We also present a new computation of the space-time structure of the wave function.…”

“…One can rewrite θ(t) and Θ(k, t) as Fourier transforms of functions with the same analyticity properties as Φ. It was shown in [6] that the functions Θ(k, t) and θ(t) have Borel summable multi-instanton expansions in 1/t valid for all t > 0, for all α, ω. These explicit formulas are essentatially obtained by pushing the Fourier contour in the lower half plane, collecting residued (resulting in small exponentials) and Hankel countours around branch cuts, which are in fact Borel sums of asymptotic series in powers of 1/t.…”

Uniformization and Constructive Analytic Continuation of Taylor Series

“…Replacing in ( 25) ω by any ϕ(ω) for any ϕ in (26) we see that ϕ * (L β F)(ω) is now analytic at both ω = 0 and ω = 1.…”