Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
Identifiers
Share
Metadata
Show full item recordDate
2011Abstract
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 ...
Palabra/s clave
Distribución asintótica a cero
Polinomios de Heine-Stieltjes
Ecuaciones diferenciales
Coeficientes polinomiales complejos
Asymptotic zero distribution
Heine-Stieltjes polynomials
Differential equations
Complex polynomial coefficients