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

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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