Asymptotic analysis of a Schr\"odinger-Poisson system with quantum wells and macroscopic nonlinearities in dimension 1

TL;DR

This paper analyzes the asymptotic behavior of a one-dimensional Schr"odinger-Poisson system with quantum wells, showing how solutions converge as the semiclassical parameter approaches zero, revealing concentrated particle densities.

Contribution

It introduces a novel asymptotic analysis of the Schr"odinger-Poisson system with quantum wells, establishing convergence results in the semiclassical limit.

Findings

Solutions converge to a nonlinear problem with concentrated density

The particle distribution remains confined within the quantum well

Unique solutions are obtained in the limit h→0

Abstract

We consider the stationary one dimensional Schr\"odinger-Poisson system on a bounded interval with a background potential describing a quantum well. Using a partition function which forces the particles to remain in the quantum well, the limit $h\rightarrow0$ in the nonlinear system leads to a uniquely solved nonlinear problem with concentrated particle density. It allows to conclude about the convergence of the solution.