Also appears in collection : 2018 - T2 - WS2 - Quantum control and feedback: foundations and applications

We consider a quantum particle in a potential V(x) and a time dependent electric field E(t), which is the control. Boscain, Caponigro, Chambrion and Sigalotti proved in [2] that, under generic assumptions on V, this PDE is approximately controllable on the L2-sphere, in sufficiently large time T. We will show that approximate controllability does not hold in arbitrarily small time. Then, we will prove several exact controllability results: small time local exact controllability by linearizing technics and quadratic obstructions to small time local exact controllability.