Bilinear control of discrete spectrum Schrödinger operators

The bilinear control problem of the Schrödinger equation $i\frac{\partial}{\partial t}ψ(t)$ $=(A+u(t) B)ψ(t)$, where $u(t)$ is the control function, is investigated through topological irreducibility of the set $\mathfrak{M}=\{e^{-it (A+u B)}, u\in \mathbb{R}, t>0\}$ 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 $B$.

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