Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to exponentially varying weights in the plane, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), based on the technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a certain matrix -problem (Its, Takhtajan), which we refer to as a soft Riemann-Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann-Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann-Hilbert problems along the interface (the free boundary for a related obstacle problem). Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real-analytic weights, the asymptotics holds in the L² sense with error in a fixed neighborhood of the closed exterior of the interface, for some constant , where . Here, m is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface (by a distance proportional to ). In particular, the zeros of the orthogonal polynomial are located in the interior of the spectral droplet, away from the droplet boundary by a distance at least proportional to .

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软黎曼-希尔伯特问题和平面正交多项式

黎曼-希尔伯特问题是全纯函数沿给定界面的跳跃问题。它们出现在各种情况下，例如，在某些非线性偏微分方程的渐近研究中以及在正交多项式的渐近分析中。 Deift 等人考虑了矩阵值 Riemann-Hilbert 问题。在 20 世纪 90 年代，采用了最速下降法的非交换适应。对于直线或圆上关于指数变化权重的正交多项式，这导致给定参数的强渐近展开。对于平面中权重呈指数变化的正交多项式，Hedenmalm 和 Wennman (2017) 基于技术上涉及的正交性不变叶化构造，获得了相应的渐近线。平面正交多项式用某个矩阵来表征-问题（Its，Takhtajan），我们将其称为软黎曼-希尔伯特问题。在这里，我们使用这个视角来提供一种简化的方法，该方法不是基于叶状结构，而是基于针对软黎曼-希尔伯特问题中的柯西势临时插入代数拟设。这允许问题沿着界面（相关障碍物问题的自由边界）分解为标量黎曼-希尔伯特问题的层次结构。受微局域分析的启发，该方法允许以这样的方式控制解：对于实分析权重，渐近保持在L ²意义上，但有误差在界面封闭外部的固定邻域中，对于某些常数， 在哪里。这里，m是多项式的次数，并且就逐点渐近而言，展开式在外部域和跨界面中主导误差项（距离与）。特别地，正交多项式的零点位于光谱液滴的内部，距液滴边界的距离至少与。