On the controllability in projections for linear quantum systems

We present sufficient conditions for the exact controllability in projection of the linear Schr{ö}dinger equations in the case where the spectrum of the free Hamiltonian is pure point. We consider the general case in which the Hamiltonian may be nonlinear with respect to the control. The controllability result applies, in particular, to Schr{ö}dinger equations with a polarizability term.