Ionization of Coulomb Systems in $${\mathbb{R}^3}$$ by Time Periodic Forcings of Arbitrary Size

Abstract

We analyze the long time behavior of solutions of the Schrodinger equation $${i\psi_t=(-\Delta-b/r+V(t,x))\psi}$$ , $${x\in\mathbb{R}^3}$$ , 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 $${2\Omega(r) \sin (\omega t-\theta)}$$ , with Ω(r) nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, i.e. $${P(t, K) = \int_K|\psi(t,x)|^2dx\to 0}$$ as t → ∞ for any compact set $${K\subset\mathbb{R}^3}$$ . 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 prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.