Equilibrium measures in the presence of weak rational external fields

TL;DR

This paper investigates equilibrium measures influenced by rational external fields created by fixed charges, focusing on the case of two negative attractors, and extends previous work on polynomial and rational external fields.

Contribution

It provides a detailed analysis of equilibrium measures under purely rational external fields without polynomial parts, addressing a complex case not previously studied.

Findings

Explicit solutions for two fixed negative charges

Extension of equilibrium measure theory to rational external fields

Simplified results despite increased mathematical difficulty

Abstract

In this paper equilibrium measures in the presence of external fields created by fixed charges are analyzed. These external fields are a particular case of the so-called rational external fields (in the sense that their derivatives are rational functions). Along with some general results, a thorough analysis of the particular case of two fixed negative charges (``attractors') is presented. As for the main tools used, this paper is a natural continuation of \cite{MOR2015}, where polynomial external fields were thoroughly studied, and \cite{OrSL2015}, where rational external fields with a polynomial part were considered. However, the absence of the polynomial part in the external fields analyzed in the current paper adds a considerable difficulty to solve the problem and justifies its separated treatment; moreover, it is noteworthy to point out the simplicity and beauty of the results obtained.