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Hyperbolic equations and its applications

双曲方程及其应用

基本信息

批准号：
02452008
负责人：
IKAWA Mitsuru
金额：
$ 2.94万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1990
资助国家：
日本
起止时间：
1990 至 1991
项目状态：
已结题

项目摘要

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.

我们研究了与双曲型方程相关的各种问题，得到了许多与双曲型方程相关的有趣结果。这些结果超出了偏微分方程理论的框架。特别地，对于波动方程在有界障碍物上的散射理论，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。针对这一问题，我们对符号流的zeta函数进行了研究。我们发展了一种方法来提取zeta函数的主要性质。这个问题与几何、代数和分析有关。我们进行了合作研究，并得到了各种有趣的结果。量子力学的散射理论，这是一个非常接近双曲问题的主题，ISOZAKI研究了多体散射问题，这一直是相当开放的，因为这个问题的难度。他介绍了一种新的方法来了解散射矩阵的精确性质。他的成果是显着的，并开辟了新的领域的数学。另一方面，问题的几何有关的偏微分方程变得非常有趣。Kasue研究员对Laplacian的谱和流形的坍缩之间的关系进行了深入的研究。通过测量行为的频谱的拉普拉斯，他清楚地表明如何顺利流形崩溃的流形不同类型。本研究是几何与解析相结合的一个典型实例。