Research on the spectrum and the monodromy related to the algebro-geometric potentials

与代数几何势相关的谱和单峰性研究

基本信息

批准号：
13640195
负责人：
OHMIYA Mayumi
金额：
$ 0.32万
依托单位：
DOSHISHA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

When the n-th Lame potential (n≧2), which is the most typical algebro-geometric potential, satisfies a kind of the degenerate condition, the Darboux-Lame equations are defined, and studied from the both of the viewpoints of isospectral deformation and isomonodromic deformation. The resulting equations are shown to be isospectral by verifying that those potentials solve the higher order KdV equation.On the one hand, those equations are shown to be isomonodromic if and only if the deformation parameter is restricted to be on some specific algebraic curve. It is also shown that the 2nd Darboux-Lame equation is transformed to the equation of Heun type with the four regular singular points by some 3-fold covering map from the underlying elliptic curve to the protective line. In addition, if the deformation parameter coincides with zero, this Heun type equation degenerates to Gauss' hypergeometric equation with the three regular singular points. By vitue of this fact and its isomonodromic property, the monodromy group associated with this Heun type equation can be concretely computed.Moreover, a kind of fibering condition of the 3-manifolds over the circle is clarified to study the fibering structure of the Jacobi variety corresponding to the spectral curve.To calculate concretely the monodromy representation, it is necessary to clarify the structure of the invariant ring associated with the finete group. An algorithm to compute the generator of the invariant ring is explored and inplemented with the computer algebra system Asir.

当第n个la脚电势（n≧2）是最典型的代数几何电位，满足了一种退化的条件时，定义了darboux-lame方程，并从等光谱变形和异构体dromicodromicodromicporormation的两个角度进行了研究。通过验证这些电势是否求解高阶KDV方程。一方面，这些方程在且仅当变形参数被限制在某些特定代数曲线上时，这些方程被证明是等异构的，这表明所得方程被证明是等谱。还表明，第二个darboux-lame方程将四个常规奇异点的方程式转换为Heun类型的方程，并通过从下面的椭圆曲线到保护线的大约3倍覆盖地图。另外，如果变形参数与零一致，则此HEUN类型方程将与三个常规奇异点的高几何方程退化为高斯的高几何方程。通过这一事实及其异构质质特性，与该HEUN类型方程相关的单型组可以进行具体计算。此外，圆圈中3个manifolds的一种光纤条件可以阐明，以研究与jacobi品种相关的classeiant concrety concriant concriant concrestions concrestiant concriant concrestions concriant concriant concriant concriant concriant concrent concrent concrient concrent controsy controsy contrys contrys contrys contromy的纤维纤维结构。精致的小组。探索并与计算机代数系统ASIR探索了一种计算不变环发生器的算法。