Concentration phenomena for the Schr\"odinger-Poisson system in $\mathbb{R}^2$

TL;DR

This paper analyzes the semiclassical limit of the Schr"odinger-Poisson system in two dimensions, revealing how solutions concentrate as the Planck constant approaches zero, with implications for quantum mechanics models.

Contribution

It provides a detailed semiclassical analysis of the planar Schr"odinger-Poisson system, identifying solution concentration phenomena as the semiclassical parameter tends to zero.

Findings

Solutions concentrate near minima of the potential V

Established existence of solution pairs in the semiclassical limit

Characterized the asymptotic behavior of solutions as epsilon approaches zero

Abstract

We perform a semiclassical analysis for the planar Schr\"odinger-Poisson system \[ \cases{ -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in $\mathbb{R}^2$},\cr -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ is a bounded external potential. We detect solution pairs $(u_\varepsilon, E_\varepsilon)$ of the system $(SP_\varepsilon)$ as~$\ge \rightarrow 0$.