  We consider the nonlinear Schrodinger equation under a partial quadratic
confinement. We show that the global dispersion corresponding to the
direction(s) with no potential is enough to prove global in time Strichartz
estimates, from which we infer the existence of wave operators thanks to
suitable vector-fields. Conversely, given an initial Cauchy datum, the solution
is global in time and asymptotically free, provided that confinement affects
one spatial direction only. This stems from anisotropic Morawetz estimates,
involving a marginal of the position density.
