Energy solution to Schr\"odinger-Poisson system in the two-dimensional whole space

TL;DR

This paper proves the global well-posedness of the two-dimensional Schr"odinger-Poisson system in the energy class despite the divergence of the Newtonian potential at infinity, using a novel decomposition approach.

Contribution

It introduces a decomposition of the nonlinearity that allows the application of perturbation methods to establish well-posedness in 2D and can be adapted to 1D cases.

Findings

Global well-posedness in 2D energy class for Schr"odinger-Poisson system

Decomposition technique for nonlinear potential handling

Applicability to one-dimensional Schr"odinger-Poisson system

Abstract

We consider the Cauchy problem of the two-dimensional Schr\"odinger-Poisson system in the energy class. Though the Newtonian potential diverges at the spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to apply the perturbation method. Our argument can be adapted to the one-dimensional problem.