Project/Area Number  02452008 
Research Category 
GrantinAid for Scientific Research (B).

Research Field 
解析学

Research Institution  Osaka University 
Principal Investigator 
IKAWA Mitsuru Osaka University Faculty of Sciences Professor, 理学部, 教授 (80028191)

CoInvestigator(Kenkyūbuntansha) 
大山 陽介 大阪大学, 理学部, 助手
川中子 正 大阪大学, 理学部, 助手 (20214661)
NAGATOMO Kiyokazu Osaka University Faculty of Sciences Lecturer, 理学部, 講師 (90172543)
TANABE Hiroki Osaka University Faculty of Sciences Professor, 理学部, 教授 (70028083)
IKEDA Nobuyuki Osaka University Faculty of Sciences Professor, 理学部, 教授 (00028078)
TSUJISHITA Tooru Osaka University Faculty of Sciences Associate Professor, 理学部, 助教授 (10107063)
KOMATSU Gen Osaka University Faculty of Sciences Associate Professor, 理学部, 助教授 (60108446)

Project Fiscal Year 
1990 – 1991

Project Status 
Completed(Fiscal Year 1991)

Budget Amount *help 
¥4,600,000 (Direct Cost : ¥4,600,000)
Fiscal Year 1991 : ¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1990 : ¥2,500,000 (Direct Cost : ¥2,500,000)

Keywords  Hyperbolic Equation / Wave Equation / Scattering Theory / Quantum Mechanics / Zeta Function / Laplacian / Spectrum / Manifold / 双曲型方程式 / 波動方程式 / 散乱理論 / 量子力学 / ゼタ関数 / ラプラシアン / スペクトル / 多様体 / 特異性の伝播 / ゼタ関数 / エルゴド性 / 幾何光学 / 漸近展開 
Research Abstract 
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 cooperatively 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.
