Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials

TL;DR

This paper studies the asymptotic distribution of zeros of Heine-Stieltjes polynomials using critical measures and quadratic differentials, revealing their limits and structural properties in complex differential equations.

Contribution

It introduces the concepts of discrete and continuous critical measures and links zero distributions to quadratic differentials, expanding understanding of polynomial solutions in complex settings.

Findings

Weak-* limits of discrete critical measures are continuous critical measures.

Zero distributions are characterized by critical measures connected to quadratic differentials.

Results include new insights into the structure of rational quadratic differentials on the Riemann sphere.

Abstract

We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest.