Bilinear control of discrete spectrum Schr\"odinger operators

The bilinear control problem of the Schr"odinger equation $i\frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi (t)$ $=(A+u(t)B)\psi (t)$, where $u(t)$ is the control function, is investigated through topological irreducibility of the set $\mathfrak{M}=\{e^{-it(A+uB)},u\in \mathbb{R},t>0\}$ of bounded operators. This allows to prove the approximate controllability of such systems when the uncontrolled Hamiltonian $A$ has a simple discrete spectrum and under an appropriate assumption on $B$.