论文标题
Riemann-Hilbert层次结构，用于硬边缘平面正交多项式
Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials
论文作者
论文摘要
我们在Jordan域$ \ Mathscr {d} $具有实时分析边界上的加权区域度量方面获得了正交多项式的完整渐近扩展。重量是固定的，并被假定为实用平稳且严格的积极，对于任何给定的精度$ \ varkappa $，扩展都具有$ \ mathrm {o}（n^{ - \ \ varkappa-1}）$在$ n $ n $依赖地区的$ n $ n $ n $ tem $ n $的$ N $中的$ N $ n $中的$ n $ tem $ n $倾向于Infriety的$ N $中的错误。主要成分是Riemann -Hilbert层次结构的推导和分析 - 标量Riemann -Hilbert问题的序列 - 这使我们能够以封闭形式表达所有高阶校正项。实际上，扩展可以理解为涉及明确操作员的诺伊曼系列。扩展定理会导致相应硬边概率波函数的半经典渐近扩展，这是在$ \ partial \ mathscr {d} $上支持的分布。
We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies - sequences of scalar Riemann-Hilbert problems - which allows us to express all higher order correction terms in closed form. In fact, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.
