Global asymptotics for polynomials orthogonal with exponential quartic weight
Global asymptotics for polynomials orthogonal with exponential quartic weight
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DOI：
10.3233/asy-2008-0937
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
R. Wong;Lun Zhang
中科院分区：
文献类型：
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作者：
R. Wong;Lun Zhang
本文研究了关于变四次权ω(x) = e - nV (x),w hereV (x) = Vt(x) = x 4 4 + t 2 x 2的正交多项式的渐近性。我们关注临界情况t =−2，在某种意义上，对于t−2，相关平衡测度的支持是一个区间，而对于t<−2，支持由两个区间组成。得到了z在三个无界区域上的全局一致渐近展开式。这些区域一起覆盖了整个复z平面。特别地，在包含原点的区域中，展开涉及与第二个Painleve方程的Hastings- McLeod解相关的Ψ函数。我们的方法是基于Deift和Zhou （Ann）引入的黎曼-希尔伯特问题的最陡下降法的改进版本。数学。137(1993),295-370)。
In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x) = e −nV (x) ,w hereV (x) = Vt(x) = x 4 4 + t 2 x 2 . We focus on the critical case t = −2, in the sense that for t −2, the support of the associated equilibrium measure is a single interval, while for t< −2, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the Ψ function affiliated with the Hastings- McLeod solution of the second Painleve equation. Our approach is based on a modified version of the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295-370).