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

本文主要研究几种凸体的散射问题。更准确地说，是经典动力学和量子力学之间的关系。该问题的重要性和难点在于，当障碍数大于或等于3时，系统将处于混沌状态。对于混沌系统，很少有研究经典力学与量子力学之间关系的著作。首先，我们研究了如何对zeta函数进行全局解析延拓，以及如何获得经典动力学中zeta函数的极点存在和不存在的信息。为此，我们试图尽可能明确地表达zeta函数。然后，对于三个障碍的情况，这是混沌系统最简单的情况。为了得到zeta函数的显式形式，我们需要知道当反射次数无限增加时，被第一和第二障碍物捕获的破碎射线的渐近行为。我们成功地得到了这种行为的非常明确的表达。利用该表达式，以第三障碍物处的反射数为参数，通过对求和进行重排，得到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一个极点。但对于高频区域，该表达式是否仍然有效，尚未得到验证。因此，这个问题是我们要研究的下一个重要对象。我们得到了被两个障碍物困住的破碎射线渐近行为的精确表达式的另一个应用。在修正拉克斯-菲利普斯猜想的研究中，一种有效的方法是使用泊松型的示踪公式。证明上述方法猜想的关键部分是对演化算子轨迹的估计。精确的表达式使得从下面对很大种类的障碍物进行估计成为可能。少

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