Minimal time for the bilinear control of Schrödinger equations

We consider a quantum particle in a potential V(x)V(x) (x∈RNx∈RN) subject to a (spatially homogeneous) time-dependent electric field E(t)E(t), which plays the role of the control. Under generic assumptions on VV, this system is approximately controllable on the L2(RN,C)L2(RN,C)-sphere, in sufficiently large times TT, 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 TT.