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

TL;DR

This paper establishes conditions for global existence and scattering versus blow-up for a 2D inhomogeneous magnetic nonlinear Schrödinger equation with Aharonov-Bohm potential, extending previous methods to non-radial data and inhomogeneous cases.

Contribution

It introduces a new approach to prove scattering for the 2D inhomogeneous magnetic NLS with Aharonov-Bohm potential, handling non-radial initial data and inhomogeneity.

Findings

Proved global existence and scattering below the ground state threshold.

Established blow-up criteria for energy solutions above the threshold.

Extended scattering results to non-radial and inhomogeneous settings.

Abstract

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).