辛矩阵上的Pfaff格
The Pfaff lattice on symplectic matrices
摘要：
Pfaff格是一个由SR-群分解产生的可积系统，类似于Toda格由QR-群分解产生。在我们最近的论文[{ \it Intern. Math. Res. Notices}，（2007）rnm 120]中，我们研究了Lax矩阵被定义为下Hessenberg矩阵的情况下的Pfaff格族。在本文中，我们处理的情况下，辛下Hessenberg Lax矩阵，这迫使Lax矩阵采取三对角的形状。然后我们证明了Pfaff格族的奇数成员是平凡的，而偶数成员等价于[Y. Kodama和J. Ye，{ \it Physica D}，{ \bf 91}（1996）321-339]。这类似于户田格族与卡茨-范·莫尔贝克系统的关系。在初始矩阵只有实特征值或虚特征值的情况下，偶流的不动点由对角为零的$2\times 2$块对角矩阵给出.我们还考虑了一族与Pfaff格有关的具有辛递归关系的斜正交多项式，发现它们可以简洁地表示为不定Toda格中的正交多项式.
英文摘要：
The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [{\it Intern. Math. Res. Notices}, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of a symplectic lower Hessenberg Lax matrix, this forces the Lax matrix to take a tridiagonal shape. We then show that the odd members of the Pfaff lattice hierarchy are trivial, while the even members are equivalent to the indefinite Toda lattice hierarchy defined in [Y. Kodama and J. Ye, {\it Physica D}, {\bf 91} (1996) 321-339]. This is analogous to the case of the Toda lattice hierarchy in the relation to the Kac-van Moerbeke system. In the case with initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by $2\times 2$ block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with symplectic recursion relation related to the Pfaff lattice, and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite Toda lattice.
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