利用Weyl m-函数研究普适性
An approach to universality using Weyl m-functions
摘要：
我们描述了一种方法的普遍性限制的正交多项式的实直线上是完全本地的，只使用边界表现的Weyl m-函数的点。我们表明，大量的普遍性的Christoffel-Darboux内核的m-函数的虚部具有正有限非切向极限的任何点。这种方法是基于研究的矩阵版本的Christoffel-Darboux内核和实现，该内核在一个点的散装普遍性是等价的事实，即相应的m-函数在同一点有正常的限制。我们的方法自动适用于其他自伴系统的$2\times 2$传输矩阵，如连续薛定谔和狄拉克运营商。对于单位圆上的正交多项式，我们也得到了类似的结果。
英文摘要：
We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel-Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with $2\times 2$ transfer matrices such as continuum Schrödinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.
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