New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

Abstract

In this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval [−1, 1], in the presence of an external fieldV. In particular, we show that ifVisC2, then the equilibrium measure is absolutely continuous with respect to Lebesgue measure, with a density which is Hölder-12 on (−1, 1), and with at worst a square root singularity at ±1. Moreover, ifVis real analytic then the support of the equilibrium measure consists of a finite number of intervals. In the cases whereV=txm,m=1, 2, 3, or 4, the equilibrium measure is computed explicitly for allt∈R. For these cases the support of the equilibrium measure consists of 1, 2, or 3 intervals, depending ontandm. We also present detailed results for the general monomial caseV=txm, for allm∈N. The regularity results for the equilibrium measure are obtained by careful analysis of the Fekete points associated to the weightenV(x)dx. The results on the support of the equilibrium measure are obtained using two different approaches: (i) an explicit formula of the kind derived by physicists for mean-field theory calculations; (ii) detailed perturbation theoretic results of the kind that are needed to analyze the zero dispersion limit of the Korteweg–de Vries equation in Lax–Levermore theory. The implications of the above results for a variety of related problems in approximation theory and the theory of orthogonal polynomials are also discussed.