Integrable Fredholm Operators and Dual Isomonodromic Deformations
Integrable Fredholm Operators and Dual Isomonodromic Deformations
复制标题
可积 Fredholm 算子和对偶等单变形
DOI：
10.1007/s002200200614
复制
发表时间：
1997
影响因子：
2.4
通讯作者：
A. Its
中科院分区：
文献类型：
--
作者：
J. Harnad;A. Its
摘要：证明了复λ-平面上一类特殊的可积积分算子K的Fredholm行列式是一类亚纯协变导数算子的同构函数族的τ-函数，在曲线段的2 m个端点上有正则奇点，在无穷远处有Poincaré指数为1的奇点.黎曼球面上相应向量丛的秩r等于定义积分核分子的指数和中不同项的数量。利用矩阵Riemann-Hilbert问题方法，导出Fredholm行列式作为Segal-Wilson和Sato意义下的τ-函数的一个恒等式，即在阿贝尔群作用方面，
在循环空间Grassmannian上的行列式线丛。协变导数算子的一个相关的对偶同构族，具有秩n= 2 m，和r个有限的正则奇点位于定义K的核的指数的值。这个家庭的变形方程是遵循从一个相关的对偶集的Riemann-Hilbert数据，其中的r指数因子的内核和它的支持的2 m端点的作用是互换。这些算子类似于积分算子，其Fredholm行列式等于K的行列式。
Abstract: The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on
the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the rôles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K.