Local existence and WKB approximation of solutions to Schr\"odinger-Poisson system in the two-dimensional whole space

TL;DR

This paper proves the local existence of solutions to the 2D Schr"odinger-Poisson system using a novel Poisson solution formula and analyzes the WKB approximation in the semiclassical limit.

Contribution

It introduces a new approach to establish local solutions and justifies the WKB approximation for the Schr"odinger-Poisson system in two dimensions.

Findings

Existence of unique local solutions in Sobolev spaces.

A new formula for solving the Poisson equation in 2D.

Validation of WKB approximation in the semiclassical limit.

Abstract

We consider the Schr\"odinger-Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schr\"odinger equation in the semiclassical limit.