On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials

TL;DR

This paper studies the asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials using the WKB method, revealing their connection to critical measures and quadratic differentials with closed trajectories.

Contribution

It introduces a new approach to derive strong asymptotics of these polynomials based on the WKB method, extending previous research.

Findings

Asymptotic formulas for Heine-Stieltjes polynomials derived

Connection established between critical measures and quadratic differentials

Enhanced understanding of polynomial solutions in complex differential equations

Abstract

We investigate the strong asymptotics of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. The solution is given in terms of critical measures (saddle points of the weighted logarithmic energy on the plane), that are tightly related to quadratic differentials with closed trajectories on the plane. The paper is a continuation of the research initiated in [arXiv:0902.0193]. However, the starting point here is the WKB method, which allows to obtain the strong asymptotics.