  We consider 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}where $\Omega(t)\subset\mathbb{R}$ is a moving domain depending
on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the
solutions of such an equation. We use the existence of a bounded reference
domain $\Omega_0$ and a specific family of unitary maps $h^\sharp(t):
L^2(\Omega(t),\mathbb{C})\longrightarrow L^2(\Omega_0,\mathbb{C})$. We show
that the conjugation by $h^\sharp$ provides a newequation of the form
\begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t)
v~~~~~\text{ on }\Omega_0\tag{$\ast\ast$} \end{equation} where
$h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian
operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an
explicit magnetic potential depending on the deformation of the domain
$\Omega(t)$. The formulation \eqref{eq_abstract2} enables to ensure the
existence of weak and strong solutions of the initial problem
\eqref{eq_abstract} on $\Omega(t)$ endowed with Dirichlet boundary conditions.
In addition, it also indicates that the correct Neumann type boundary
conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic
ones$$\partial_\nu u(t)+i\langle\nu| A\rangle u(t)=0,$$even though
\eqref{eq_abstract} has no magnetic term. All the previous results are also
studied in presence of diffusion coefficients as well as magnetic and electric
potentials. Finally, we prove some associated byproducts as an adiabatic result
for slow deformations of the domain and atime-dependent version of the
so-called ``Moser's trick''. We use this outcome in order to simplify Equation
\eqref{eq_abstract2} and to guarantee the well-posedness for slightly less
regular deformations of $\Omega(t)$.
