coupled for some $s>0$. If the [Lie–Galerkin Control Condition]{} holds then system  is approximately simultaneously controllable for the $s/2$-norm.

Example: the infinite potential well {#sec:buca}
------------------------------------

We present the case of a particle confined in the interval $(-1/2,1/2)$ as a toy model to compare the result in [@ancoranoi] and Theorem \[metalemma-tracking\] on a single-input system. The model has been extensively studied by several authors in the last decade and it has been the first quantum system for which a positive controllability result has been obtained (see [@beauchard-coron]). In [@ancoranoi] an approximate simultaneous controllability has been obtained with geometric methods and using perturbations techniques. Indeed this model presents several resonances preventing the direct application of the results in [@ancoranoi].

The Schrödinger equation writes $$\label{EQ_potential_well}
 i \frac{\partial \psi}{\partial t}=-\frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} - u(t) x \psi(x,t)$$ with the boundary conditions $\psi(-1/2,t)=\psi(1/2,t)=0$ for every $t \in {{\mathbb{R}}}$. We consider controls $u(\cdot)$ piecewise constant with values in $U=[-\delta,\delta]$ for some $\delta>0$.

In this case ${\mathcal{H}}=L^2 \left ( (-1/2,1/2), {{\mathbb{C}}}\right )$ endowed with the Hermitian product $\langle \psi_1,\psi_2\rangle= \int_{-1/2}^{1/2} \overline{\psi_1(x)} \psi_2(x) dx$. The operators $A$ and $B=B_{1}$ are defined by $A\psi= i\frac{1}{2} \frac{\partial^2 \psi}{\partial x^2}$ for every $\psi$ in $D(A)= (H_2 \cap  H_0^1 )\left((-1/2,1/2), {{\mathbb{C}}}\right)$ 