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 $i{\psi }_t=(-\Delta -b/r+V(t,x))\psi$, $x\in\RR^3$, $r=|x|$, describing a Coulomb system subjected to a spatially compactly supported time periodic potential $V(t,x)=V(t+2\pi /\omega ,x)$ with zero time average. We show that, for any $V(t,x)$ of the form $2\Omega (r)\sin (\omega t-\theta )$, with $\Omega (r)$ nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, {\em i.e.} $P(t,K)={\int }_K|\psi (t,x){|}^{2}dx\to 0$ 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.

References in corpus (1)

• Time asymptotics of the Schroedinger wave function in time-periodic potentials