Interaction Morawetz estimate for the magnetic Schrödinger equation and applications

Abstract

We establish an interaction Morawetz estimate for the magnetic Schrödinger 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 Schrödinger equation for $n = 3.$

23 pages; Added some remarks in the introduction and more details in the proof of LWP and GWP, fixed typos. To appear in DIE