Hyperbolic equations and its applications

我们学习了与双曲方程有关的各种学科，得到了许多与双曲方程有关的有趣结果。这些结果超出了偏微分方程理论的范围。特别是关于有界障碍物对波动方程的散射理论，我们明确了散射矩阵的基本性质与动力系统在障碍物外部的zeta函数密切相关。针对这个问题，我们对符号流的zeta函数进行了研究。我们提出了一种方法来提出函数的主要性质。这个问题与几何、代数和分析有关。我们合作研究，得到了各种有趣的结果。量子力学中的散射理论，是一个非常接近双曲问题的学科，ISOZAKI研究了多体散射问题，由于问题的艰难性，这一问题一直保持着相当开放的状态。他介绍了一种新的方法来了解散射矩阵的精确性质。他的成果是显著的，并开辟了新的数学领域。另一方面，与偏微分方程相关的几何问题变得非常有趣。研究者Kasue对拉普拉斯谱与流形坍缩之间的关系进行了深入的研究。通过测量拉普拉斯谱的行为，他明确了光滑流形如何坍缩成不同类型的流形。本研究是几何与分析相结合的典型案例。

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.