to the Euclidean coordinates through the identities $$x = \sin \theta \cos \varphi, \quad y = \sin \theta\sin \varphi,\quad z = \cos \theta,$$ while $\Delta$ is the Laplace–Beltrami operator on the sphere $S^2$ (called in this context the *angular momentum operator*), i.e., $$\Delta = \frac{1}{\sin \theta} {\frac{\partial}{\partial \theta}} \left( \sin \theta {\frac{\partial}{\partial \theta}}\right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \varphi^2}.$$

The wavefunction $\psi$ evolves in the unit sphere ${\cal S}$ of ${\cal H}=L^2(S^2,{{\mathbb{C}}})$.

As a consequence of the general multi-input result presented in Section \[sec:mainresults\] we have that  is approximately controllable with arbitrarily small controls. A stronger statement, including simultaneous controllability in $H^{s}$ and stalking, is given in Section \[rotational\].

\[controllability\] For every $\psi^0$, $\psi^1$ belonging to ${\cal S}$ and every $\delta>0$, there exist $T>0$ and $u\in L^\infty([0,T],[0,\delta]^{3})$, such that the solution $\psi(\cdot)$ of equation , corresponding to the control $u$ and with initial condition $\psi_0$, satisfies $\|\psi^1-\psi(T)\|<{\varepsilon}$.

There are two main difficulties preventing the application to this system results previously in the literature. Firstly, we deal here with several control parameters, while those general results were specifically conceived for the single-input case. Notice that, because of symmetry obstructions, equation  is not controllable with only two of the three controls $u_1$, $u_2$, 