  We review different properties related to the Cauchy problem for the
(nonlinear) Schrodinger equation with a smooth potential. For
energy-subcritical nonlinearities and at most quadratic potentials, we
investigate the necessary decay in space in order for the Cauchy problem to be
locally (and globally) well-posed. The characterization of the minimal decay is
different in the case of super-quadratic potentials.
