Hyperbolic equations and its applications

我们研究了与双曲型方程相关的各种课题，得到了许多与双曲型方程相关的有趣的结果。这些结果超出了偏微分方程组理论的框架。特别是，关于有界障碍物对波动方程的散射理论，我们阐明了散射矩阵的基本性质与障碍物外动力系统的Zeta函数密切相关。针对这个问题，我们研究了符号流的Zeta函数。我们开发了一种方法来提取Zeta函数的主要性质。这道题与几何、代数和分析有关。我们进行了合作研究，得到了许多有趣的结果。量子力学的散射理论是一个非常接近双曲问题的学科，Isozaki对多体散射问题进行了研究，由于问题的难度，这个问题一直是相当开放的。他介绍了一种新的方法来了解散射矩阵的精确性质。另一方面，与偏微分方程有关的几何问题也变得非常有趣。研究人员Kasue对拉普拉斯谱和流形崩溃之间的关系进行了深入研究。通过测量拉普拉斯流形的谱行为，他阐明了光滑流形如何折叠成不同类型的流形。本研究是几何与分析相结合的典型例子。

We studied the various subjects related to hyperbolic equations, and we get many interesting results related to hyperbolic equations. These results are beyond the frame of the theory of partial differential equations. Especially, concerning to the scattering theory for the wave equation by bounded obstacles, we made clear that the fundamental properties of scattering matrices closely related to the zeta functions of dynamical system in the outside of obstacles. As to this problem, we made studies on the zeta functions of symbolic flows. We developed the method to take out the main properties of the zeta functions. This problem has relations with geometry, algebra and analysis. We made researches co-operatively and got various interesting results.Scattering theory of quantum mechanics, which is a subject very close to hyperbolic problems, ISOZAKI made study on the scattering of many body problem, which had been remained quite open, because the difficulty of the problem. He introduced a new method to know the precise properties of scattering matrices. His results are remarkable and opened new fields of mathematics.On the other hand, the problems of geometrics related to partial differential equations became very interesting. Investigator Kasue made deep studies on the relationships between the spectrum of the Laplacian and the collapse of manifolds. By measuring the behavior of spectrum of the Laplacian he made clear how smooth manifolds collapse to manifolds of different type. This research is a typical example combining the geometry and analysis.