Integrable geodesic flows and semi-classical approximations

可积测地线流和半经典近似

基本信息

批准号：
11640053
负责人：
KIYOHARA Kazuyoshi
金额：
$ 2.3万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

We investigated the structures of Kahler-Liouville manifolds which are not necessarily of type (A). As a consequence, we showed that every compact, proper Kahler-Liouville manifold has a bundle structure such that the fiber is a Kahler-Liouville manifold whose geodesic flow is integrable, and the base is (locally) a product of one-dimensional Kahler manifolds. The fiber naturally possesses the structure of toric variety. Also, we obtain another class, called of type (B), of Kahler-Liouville manifolds whose geodesic flows are integrable.Also, we obtained many examples of "Hermite-Liouville" manifolds whose geodesic flows are integrable. The first examples are those defined over Hopf surfaces. Others are those defined over complex projective spaces. In the latter case, the idea of construction is as follows. We use two structures of real Liouville manifolds on the real projective space. One of them is used to prepare the "frame of complexification", while the other is used to produce a comlexified metric and first integrals. If those two Liouville structures are the same, then the result becomes Kahler-Liouville manifold.Moreover, 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 them. As a result, we found that this method gives new approximations when the corresponding invariant tori tend to a critical one.

我们研究了不一定为（a）类型的卡勒·洛维尔流形的结构。结果，我们表明，每个紧凑的，适当的Kahler-Liouville歧管都有一个捆扎结构，因此纤维是Kahler-Liouville的歧管，其地理流量是可集成的，并且基部是（本地）一维Kahler歧管的产物。纤维自然具有复曲面的结构。另外，我们获得了Kahler-Liouville歧管的另一个称为（b）的类别，其大地歧管是可集成的。此外，我们获得了许多“ Hermite-liouville”歧管的示例，其地理流是可集成的。第一个示例是在HOPF表面上定义的示例。其他是在复杂的投影空间上定义的。在后一种情况下，施工的想法如下。我们在真实的投影空间上使用了真正的liouville歧管结构。其中一个用于准备“络合框架”，而另一个用于生成共同体的度量和第一积分。如果这两个liouville结构相同，那么结果就变成了Kahler-Liouville歧管。此外，我们研究了Liouville表面上的Laplacian的光谱，以差异为2-Sphere。我们将本征函数的定义方程式分解为圆圈上的一对普通微分方程，并将“半经典近似”应用于它们。结果，我们发现，当相应的不变托里趋向于关键时，这种方法给出了新的近似值。