Ionization of Coulomb systems in $\RR^3$ by time periodic forcings of arbitrary size

Abstract

We analyze the long time behavior of solutions of the Schrödinger equation $iψ_t=(-Δ-b/r+V(t,x))ψ$, $x\in\RR^3$, $r=|x|$, describing a Coulomb system subjected to a spatially compactly supported time periodic potential $V(t,x)=V(t+2π/ω,x)$ with zero time average. We show that, for any $V(t,x)$ of the form $2Ω(r)\sin (ωt-θ)$, with $Ω(r)$ nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, {\em i.e.} $P(t,K)=\int_K|ψ(t,x)|^2dx\to 0$ as $t\to\infty$ for any compact set $K\subset\RR^3$. Furthermore, if the initial state is compactly supported and has only finitely many spherical harmonic modes, then $P(t,K)$ decays like $t^{-5/3}$ as $t \to \infty $. To prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.