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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 adichotomy of global existence and scattering versus blow-up of energy solutionsunder the ground state threshold in the inter-critical regime. The scatteringis obtained by using the new approach of Dodson-Murphy (A new proof ofscattering 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 andMorawetz estimates. The novelty here is twice: we investigate the case$\varrho\alpha\neq0$ and we consider general energy initial data (notnecessarily radially symmetric).

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential