Classical theory and quantization on integrable geodesic flows

可积测地流的经典理论和量化

基本信息

批准号：
09640082
负责人：
KIYOHARA Kazuyoshi
金额：
$ 1.92万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

For "classical theory" we have obtained two major results. One of them may be stated as follows : Let M be a riemannian manifold diffeomorphic to 2-sphere, and let F be a first integral of its geodesic flow that is a polynomial of degree kappa on each fiber. Such a pair (M, F) is well-understood if kappa = 1,2. When kappa <greater than or equal> 3, however, no nontrivial example has been known except one-parameter families for kappa 3 and 4. In this research we constructed families of such (M, F) (parametrized by functions in one variable) for every kappa <greater than or equal> 3. Moreover, we proved that they are C_<2pi>-manifolds, i.e., every geodesic is closed and has the common length 2pi. The other result is a construction of "Hermite-Liouville structure" on Hopf surfaces. The idea is analogous to Kahler-Liouville manifold, which is a complexifled version of Liouville manifold established by the head investigator. Despite the significance of the Kahler condition in the whole theory of Kahler-Liouville manifolds, this result seems to suggest the existence of another complexification scheme for LiouvilFor "quantization" we studied spectra of the laplacian on Liouville surfaces diffeomorphic to 2-sphere. We decomposed the defining equation of the eigenfunctions into a pair of ordinary differential equations on circles, and applied semiclassical approximation to each of them. As a result, we found that this method gives a nice approximation when the corresponding invariant tori are sufficiently close to a critical one, as well as the case where the tori are located far from the critical ones. We think this result will be more refined.

对于“经典理论”我们得到了两个主要结果。其中一个可以表述如下：令 M 为与 2-球体微分同胚的黎曼流形，并令 F 为其测地线流的第一积分，该测地线流是每个纤维上 kappa 次的多项式。如果 kappa = 1,2，这样的对 (M, F) 就很好理解了。然而，当 kappa <大于或等于> 3 时，除了 kappa 3 和 4 的单参数族外，没有已知的重要示例。在这项研究中，我们构建了此类 (M, F) 的族（通过一个变量中的函数进行参数化）对于每个 kappa <大于或等于> 3。此外，我们证明它们是 C_<2pi>-流形，即每个测地线都是闭合的并且具有公共长度2pi。另一个结果是在 Hopf 表面上构建“Hermite-Liouville 结构”。这个想法类似于卡勒-刘维尔流形，它是首席研究员建立的刘维尔流形的复杂版本。尽管卡勒条件在整个卡勒-刘维尔流形理论中具有重要意义，但这一结果似乎表明存在另一种刘维尔复化方案。对于“量子化”，我们研究了微分同胚到 2 球体的刘维尔表面上的拉普拉斯谱。我们将特征函数的定义方程分解为一对圆上的常微分方程，并对每个方程应用半经典近似。结果，我们发现，当相应的不变环面足够接近临界环面时，以及环面远离关键环面的情况时，该方法给出了很好的近似。我们认为这个结果会更加精致。