Minimal time for the approximate bilinear control of Schrödinger equations

We consider a quantum particle in a potential V ( x ) subject to a time‐dependent (and spatially homogeneous) electric field E ( t ) (the control). Boscain, Caponigro, Chambrion, and Sigalotti proved that, under generic assumptions on V , this system is approximately controllable on the L 2 ( R N , C ) unit sphere, in sufficiently large time T . In the present article, we show that, for a large class of initial states (dense in L 2 ( R N , C ) unit sphere), approximate controllability does not hold in arbitrarily small time. This generalizes our previous result for Gaussian initial conditions. Furthermore, we prove that the minimal time can in fact be arbitrarily large.