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New results on the equilibrium measure for logarithmic potentials in the presence of an external field. (English) Zbl 0918.31001
(Authors’ summary) The authors 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 field \(V\). In particular, they show that if \(V\) is \(C^2\), then the equilibrium measure is absolutely continuous with respect to Lebesgue measure, with a density which is Hölder-\({1\over 2}\) on \((-1,1)\), and with at worst a square root singularity at \(\pm 1\). Moreover, if \(V\) is real analytic then the support of the equilibrium measure consists of a finite number of intervals.
In the case where \(V= tx^m\), \(m= 1,2,3\), or 4, the equilibrium measure is computed explicitly for all \(t\in\mathbb{R}\). For these cases the support of the equilibrium measure consists of 1, 2, or 3 intervals, depending on \(t\) and \(m\). The authors also present detailed results for the general monomial case \(V= tx^m\), for all \(m\in\mathbb{N}\).
The regularity results for the equilibrium measure are obtained by careful analysis of the Fekete points associated to the weight \(e^{nV(x)}dx\). 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.

MSC:
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31A35 Connections of harmonic functions with differential equations in two dimensions
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
35Q58 Other completely integrable PDE (MSC2000)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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