Averaging of strong magnetic nonlinear Schr\"odinger equations in the energy space

In this study, we consider two nonlinear Schr\"{o}dinger-type models that are derived by R L. Frank, F. M\'{e}hats, C. Sparber [arXiv:1611.01574] to study 3D nonlinear Schr\"{o}dinger equations under strong magnetic fields. One model is derived by spatial scaling and the other is obtained by averaging the spatial scaled model over time. We study these models in the energy space to obtain global solutions and improve the convergence result over an arbitrarily long time. Regarding the nonic nonlinear power of the time averaged model, we prove a scattering result under a scaling-invariant small-energy condition, which underlines energy-criticality of the nonic case.