  This paper studies the local exact controllability and the local
stabilization of the semilinear Schr\"odinger equation posed on a product of
$n$ intervals ($n\ge 1$). Both internal and boundary controls are considered,
and the results are given with periodic (resp. Dirichlet or Neumann) boundary
conditions. In the case of internal control, we obtain local controllability
results which are sharp as far as the localization of the control region and
the smoothness of the state space are concerned. It is also proved that for the
linear Schr\"odinger equation with Dirichlet control, the exact controllability
holds in $H^{-1}(\Omega)$ whenever the control region contains a neighborhood
of a vertex.
