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Study on the relationships between the classical mechanics and the chaotic properties of wave motions

经典力学与波动混沌特性关系研究

基本信息

批准号：
12440047
负责人：
IKAWA Mitsuru
金额：
$ 3.9万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440047/
关键词：
clssical dynamics Ruelle operator chaos zeta function meromorphic pole 有利型波動

项目摘要

This research mainly concerns with the scattering by several convex bodies. More precisely, the relationships between the classical dynamics and the quantum mechanics. The importance and the difficulties of this problem come from the fact that, when the number of obstacle is greater than or equal 3, the system becomes chaotic. Concerning chaotic systems, there are few works on the relationships between classical and quantum mechanics.First we studied how we can make globally the analytic continuation of the zeta functions, and how we can get informations on existence and non-existence of poles of the dynamical zeta functions of the classical dynamics.To this end, we tried to express the zeta function as explicitly as possible. Then, for the case of three obstacles which is the simplest case of chaotic systems. Under this situation, we made a more assumption that the third obstacle is small comparing to the others, To get an explicit form of the zeta function, it is necessary to know th … More e asymptotic behavior of broken rays trapped by the first and second obstacles when the number of reflections increases infinitely. We succeeded to get very explicit expression of the behavior. Using this expression, treating the number of reflection at the third obstacle as a parameter, we get an explicit expression of the zeta function by making rearrangement of the summation. This expression enables us to find a pole in the region of low frequency. But it is not verified that this expression is still valid for the region of high frequencies. Thus, this problem is the next important object we have to study.We have another application of the precise expression of asymptotic behavior of broken rays trapped by two obstacles. In the study of the modified Lax-Phillips conjecture, one efficient method is to use the trace formula of Poisson type. Crucial part of the proof of the conjecture of the above method is an estimate from the below of the trace of the evolution operator. The precise expression make possible to get an estimate from below for very wide class of obstacles. Less

本文主要研究多个凸体的散射问题。更准确地说，是经典动力学和量子力学之间的关系。这一问题的重要性和困难性在于当障碍物的个数大于或等于3时，系统将变得混沌。关于混沌系统，经典力学与量子力学之间的关系研究较少，本文首先研究了如何使zeta函数在全局上解析延拓，以及如何得到经典动力学zeta函数极点的存在性和不存在性的信息，并试图尽可能明确地表示zeta函数.然后，对于三个障碍物的情况，这是混沌系统的最简单的情况。在这种情况下，我们做了一个更小的假设，即第三个障碍物相对于其他障碍物较小，为了得到zeta函数的显式形式，需要知道第三个障碍物的位置， ...更多信息当反射次数无限增加时，第一和第二障碍物捕获的破碎光线的渐近行为。我们成功地得到了行为的非常明确的表达。利用该表达式，将第三障碍物处的反射次数作为参数，通过对求和进行重排，得到了zeta函数的显式表达式。这个表达式使我们能够在低频区域找到极点。但这一表达式在高频区是否仍然有效，尚未得到证实。因此，这个问题是我们要研究的下一个重要课题。我们得到了被两个障碍物俘获的破碎光线的渐近行为的精确表达式的另一个应用。在修正的Lax-Phillips猜想的研究中，一种有效的方法是利用Poisson型迹公式。证明上述方法的猜想的关键部分是从下面的演化算子的迹的估计。精确的表达式使得可以从下面获得对非常广泛的障碍物类别的估计。少