Interaction Morawetz estimate for the magnetic Schr\"odinger equation and applications

We establish an interaction Morawetz estimate for the magnetic Schr"odinger equation for $n\ge 3$ under certain smallness conditions on the gauge potentials, but with almost optimal decay. As an application, we prove global wellposedness and scattering in $H^1$ for the cubic defocusing magnetic Schr"odinger equation for $n=3.$