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

We consider the magnetic nonlinear inhomogeneous Schrödinger equation $$i{\mathrm{\partial }}_{t}u-{(-i\mathrm{\nabla }+\frac{\alpha }{\mathrm{\mid }x{\mathrm{\mid }}^2}(-x_2,x_1))}^{2}u=\pm \mathrm{\mid }x{\mathrm{\mid }}^{-\varrho }\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^2,$$ 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 $\varrho \alpha \mathrm{\ne }0$ and we consider general energy initial data (not necessarily radially symmetric).