On a class of equilibrium problems in the real axis

[摘要] In a series of seminal papers, Thomas J. Stieltjes (1856-1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lame equations), the so-called Heine-Stieltjes polynomials. In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x(1),..., x(n) is an element of R are in presence of a finite family of attractors (i.e., negative charges) z(1),..., z(m) is an element of C\R, is considered and its connection with certain class of Lame-type equations is shown. In addition, we study the situation when both n -> infinity and m -> infinity, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields. (C) 2010 Elsevier B.V. All rights reserved.

[发布日期] 2010-12-15

[效力级别]  Proceedings Paper

[关键词] Heine-Stieltjes polynomials;Lame equation;Equilibrium measures;External fields