Control of the Schr\"odinger equation by slow deformations of the domain

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 $\mathrm{\Omega }(t)\subset {\mathbb{R}}^N$ is a time-varying domain. We prove the global approximate controllability of \eqref{eq_abstract} in $L^2(\mathrm{\Omega })$, via an adiabatic deformation $\mathrm{\Omega }(t)\subset \mathbb{R}$ ($t\in [0,T]$) such that $\mathrm{\Omega }(0)=\mathrm{\Omega }(T)=\mathrm{\Omega }$. 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.