Exact results for ionization of model atomic systems

Abstract

We review recent rigorous results concerning the ionization of model quantum systems by time-periodic external fields. The systems we consider consist of a single particle (electron) with a reference Hamiltonian H0=−Δ+V0(x) (x∊Rd) having both bound and continuum states. Starting from an initially localized state ψ0(x)∊L2(Rd), the system is subjected for t≥0 to an arbitrary strength time-periodic potential V1(x,t)=V1(x,t+2π/ω). We prove that for a large class of V0(x) and V1(x,t), the wave function ψ(x,t) will delocalize as t→∞, i.e., the system will ionize. The only exceptions are cases where there are time-periodic bound states of the Floquet operator associated with H0+V1. These do occur (albeit rarely) when V1 is not small. For spatially rapidly decaying V0 and V1, ψ(x,t) is generally given, for very long times, by a power series in t−1/2 which we prove in some cases to be Borel summable. For the Coulomb potential V0(x)=−b|x|−1 in R3, we prove ionization for V1(x,t)=V1(|x|)sin(ωt−θ), V1(|x|)=0 for |x|>R and V1(x)>0 for |x|≤R. For this model, if ψ0 is compactly supported both in x and in angular momentum, L, we obtain that ψ(x,t)∼O(t−5/6) as t→∞.