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

We consider nonlinear Schr\"odinger equation with strong magnetic fields in 3-dimension. To analyze this, R L. Frank, F. M\'ehats, C. Sparber (2017) derived two nonlinear Schr\"odinger type models. One model is derived by spatial scaling and the other model is obtained by averaging the spatial scaled model in time. We study these models in energy space to obtain global solutions and improve the convergence result to arbitrarily long time. In the case of nonic nonlinear power of the time averaged model, we prove a scattering result under a scaling-invariant small-energy condition, which indicates energy-criticality of the nonic case.Comment: 34 pages. arXiv admin note: text overlap with arXiv:1611.01574 by other author