1991 Fiscal Year Final Research Report Summary
Hyperbolic equations and its applications
| Project/Area Number |
02452008
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Osaka University |
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Principal Investigator |
IKAWA Mitsuru Osaka University Faculty of Sciences Professor, 理学部, 教授 (80028191)
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| Co-Investigator(Kenkyū-buntansha) |
NAGATOMO Kiyokazu Osaka University Faculty of Sciences Lecturer, 理学部, 講師 (90172543)
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)
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)
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| Project Period (FY) |
1990 – 1991
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| Keywords | Hyperbolic Equation / Wave Equation / Scattering Theory / Quantum Mechanics / Zeta Function / Laplacian / Spectrum / Manifold |
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| 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 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.
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