Riemann曲面上的Padé逼近与KP τ函数
Padé\ approximants on Riemann surfaces and KP tau functions
摘要：
本文有两个相对不同的，但连接的目标;第一个是定义的概念Padé\近似的Weyl-Stiltjes变换的任意紧致黎曼曲面的高属。该数据由黎曼曲面中的轮廓线和其上的测量值以及点附近的局部坐标的附加数据和度因子$g$组成。由此产生的Padé-样近似的分解子也满足正交关系，并且是适当的线丛的部分。一个黎曼-希尔伯特问题的秩二方阵的特征，这些正交部分，在一个类似的方式，普通的正交多项式的情况下。第二部分扩展了这一思想，探讨其与可积系统的联系。相同的数据可用于定义两个线束序列之间的配对。在变形空间中的轨迹，配对成为退化为固定的程度与“tau”函数的零点重合。我们展示了这个tau函数如何满足Kadomtsev-Petviashvili层次结构相对于任何变形参数，和一定的修改2-户田层次结构时，考虑整个序列的tau函数。我们还展示了如何建设有关的Krichever建设代数几何解决方案。
英文摘要：
The paper has two relatively distinct but connected goals; the first is to define the notion of Padé\ approximation of Weyl-Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree $g$. The denominators of the resulting Padé--like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann--Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a "tau" function. We show how this tau function satisfies the Kadomtsev--Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2--Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro--geometric solutions.
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