We consider the magnetic nonlinear inhomogeneous Schr\"odinger equation $$i\partial_t u -\left(-i\nabla+\frac{ \alpha}{ |x|^2}(-x_2,x_1)\right)^2 u =\pm|x|^{ -\varrho}|u|^{ p-1}u,\quad (t,x)\in \mathbb{ R}\times \mathbb{ R}^2,$$ where $\alpha\in\mathbb{ R}\setminus\mathbb{ Z},\,\varrho>0,\,p>1$. We prove a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground state threshold in the inter-critical regime. The scattering is obtained by using the new approach of Dodson-Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, { Proc. Am. Math. Soc.} (2017)). This method is based on Tao's scattering criteria and Morawetz estimates. The novelty here is twice: we investigate the case $\varrho\alpha\neq0$ and we consider general energy initial data (not necessarily radially symmetric).
Categories: math.AP math-ph math.MP