  The aim of this work is to study the controllability of the Schr\"odinger
equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta
u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation} with Dirichlet
boundary conditions, where $\Omega(t)\subset\mathbb{R}^N$ is a time-varying
domain. We prove the global approximate controllability of \eqref{eq_abstract}
in $L^2(\Omega)$, via an adiabatic deformation $\Omega(t)\subset\mathbb{R}$
($t\in[0,T]$) such that $\Omega(0)=\Omega(T)=\Omega$. This control is strongly
based on the Hamiltonian structure of \eqref{eq_abstract} provided by [18],
which enables the use of adiabatic motions. We also discuss several explicit
interesting controls that we perform in the specific framework of rectangular
domains.
