High density limit of the stationary one dimensional Schr\"odinger-Poisson system

TL;DR

This paper investigates the behavior of the one-dimensional Schr"odinger-Poisson system in the high-density limit, revealing that only the lowest energy state is occupied and characterizing the limiting electrostatic potential.

Contribution

It introduces a convex minimization approach to analyze the system and demonstrates the asymptotic occupation of the first energy level in the small Debye length limit.

Findings

Only the first energy level is asymptotically occupied.

Electrostatic potential converges to a boundary layer profile.

The boundary layer potential is computed via a half-space Schr"odinger-Poisson system.

Abstract

The stationary one dimensional Schr\"odinger-Poisson system on a bounded interval is considered in the limit of a small Debye length (or small temperature). Electrons are supposed to be in a mixed state with the Boltzmann statistics. Using various reformulations of the system as convex minimization problems, we show that only the first energy level is asymptotically occupied. The electrostatic potential is shown to converge towards a boundary layer potential with a profile computed by means of a half space Schr\"odinger-Poisson system.