Minimal time for the bilinear control of Schrِdinger equations

We consider a quantum particle in a potential V ( x ) ( x ∈ R N ) subject to a (spatially homogeneous) time-dependent electric field E ( t ) , which plays the role of the control. Under generic assumptions on V , this system is approximately controllable on the L 2 ( R N , C ) -sphere, in sufficiently large times T , as proved by Boscain, Caponigro, Chambrion and Sigalotti (2012). In the present article, we show that this approximate controllability result is false in small time. As a consequence, the result by Boscain et al. is, in some sense, optimal with respect to the control time T .