A Kohn-Sham system at zero temperature

TL;DR

This paper analyzes a one-dimensional zero-temperature Kohn-Sham system for spin particles, proving existence of solutions, deriving estimates, and exploring behavior as temperature approaches zero.

Contribution

It establishes existence, a priori estimates, and uniqueness results for the zero-temperature Kohn-Sham system, extending understanding of such systems in semiconductor nanostructures.

Findings

Existence of solutions proven using Schauder's fixed point theorem.

A priori estimates for eigenvalues and particle density established.

Behavior of the system as temperature approaches zero analyzed.

Abstract

An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor {nano}structures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schr\"odinger operator with effective Kohn-Sham potential and obtain $W^{1,2}$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.