New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field
New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field
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DOI：
10.1006/jath.1997.3229
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发表时间：
1998-12
影响因子：
0.9
通讯作者：
P. Deift;T. Kriecherbauer;K. Mclaughlin
中科院分区：
文献类型：
--
作者：
P. Deift;T. Kriecherbauer;K. Mclaughlin
在本文中，我们使用的技术，从理论的常微分方程，也从逆散射理论，以获得各种结果的规律性和支持性质的平衡措施的对数势的有限区间？1，1]，在存在一个外部场V。特别地，我们证明了如果VisC ~ 2，则平衡测度关于Lebesgue测度是绝对连续的，其密度在（？）1，1），并且在最坏的情况下，平方根奇异性为±1。此外，如果V是真实的解析的，则平衡测度的支撑由有限个区间组成。在V =txm，m=1，2，3，或4的情况下，平衡措施计算明确的所有t？R.对于这些情况下的支持的平衡措施包括1，2，或3个区间，取决于ontandm。我们还提出了详细的结果，一般单项caseV=txm，allm？N.平衡测度的正则性结果是通过仔细分析与加权V（x）dx相关的Fekete点得到的。的平衡措施的支持上的结果是使用两种不同的方法获得：（i）一个明确的公式的物理学家推导出的平均场理论计算的那种;（ii）详细的微扰理论的结果，需要分析的零色散极限的Korteweg？弗里斯方程在拉克斯？Levermore理论上述结果的各种相关问题的逼近理论和正交多项式理论的影响也进行了讨论。
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 Holder-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.