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

Percy Deift, T. Kriecherbauer, K. T R McLaughlin

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)388-475
Number of pages88
JournalJournal of Approximation Theory
Volume95
Issue number3
DOIs
StatePublished - Dec 1998

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Korteweg-de Vries equation
Logarithmic Potential
Approximation theory
Equilibrium Measure
Mean field theory
External Field
Polynomials
Scattering
Interval
Fekete Points
Regularity
Approximation Theory
Scattering Theory
Inverse Scattering
Mean-field Theory
Monomial
Lebesgue Measure
Absolutely Continuous
Korteweg-de Vries Equation
Square root

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics
  • Numerical Analysis

Cite this

New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field. / Deift, Percy; Kriecherbauer, T.; McLaughlin, K. T R.

In: Journal of Approximation Theory, Vol. 95, No. 3, 12.1998, p. 388-475.

Research output: Contribution to journalArticle

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