On a class of equilibrium problems in the real axis
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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 (Lamé equations), the so-called Heine–Stieltjes polynomials.In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x1,…,xn∈R are in presence of a finite family of “attractors” (i.e., negative charges) z1,…,zm∈C∖R, is considered and its connection with certain class of Lamé-type equations is shown. In addition, we study the situation when both n→∞ and m→∞, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.
论文关键词:Heine–Stieltjes polynomials,Lamé equation,Equilibrium measures,External fields
论文评审过程:Received 16 September 2009, Revised 5 April 2010, Available online 25 May 2010.
论文官网地址:https://doi.org/10.1016/j.cam.2010.05.027