Minimal energy point systems on the unit circle and the real line

TL;DR

This paper analyzes the equilibrium configurations of electrons and protons on the unit circle under logarithmic energy, proving all critical points are global minima and that any electron configuration is stable in an external proton field.

Contribution

It demonstrates that all critical points of the discrete logarithmic energy are global minima and establishes the stability of any electron configuration under an external proton field.

Findings

All critical points are global minima.

Configurations of electrons are stable in external fields.

Refinement of a recent stability result.

Abstract

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons.