Controllability properties of discrete-spectrum Schrödinger equations

We state an approximate controllability result for the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. This result applies both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. In addition we get some controllability properties for the density matrix. Finally we show, by means of specific examples, how these results can be applied.