Interaction Morawetz estimate for the magnetic Schr\"odinger equation   and applications
  We establish an interaction Morawetz estimate for the magnetic Schr\"odinger equation for $n\geq 3$ 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 $n = 3.$ 
James Colliander, Magdalena Czubak, Jeonghun Lee
math.AP