2003 Fiscal Year Final Research Report Summary
Mathematical analysis of scattering phenomena and inverse problems
| Project/Area Number |
13440048
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
ISOZAKI Hiroshi Tokyo Metropolitan University, Graduate School of Science, Professor, 理学研究科, 教授 (90111913)
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| Co-Investigator(Kenkyū-buntansha) |
MOCHIZUKI Kiyoshi Chuo University, Department of Science and Engineering, Professor, 理工学部, 教授 (80026773)
吉富 和志 東京都立大学, 理学研究科, 助教授 (40304729)
OKADA Masami Tokyo Metropolitan University, Graduate School of Science, Associate Professor, 理学研究科, 教授 (00152314)
TAMURA Hideo Okayama University, Department of Mathematics, Professor, 理学部, 教授 (30022734)
NAKAMURA Gem Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50118535)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Scatteirng theory / Inverse problems / Dirichlet-Neumann map / Schrodinger operators / Hyperbolic space / S matrix / Electric Impedance Tomography / Numerical analysis |
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| Research Abstract |
This reserach project aimed at the development of the study of inverse problems arising from the mathematical analysis of scattering phenoma. The main theme was the study of spectra of differential operators. As a main conference project, we have organized an international workshop of inverse problems on October 2002 at Kyoto, where leading reserchers of this field came together, and promoted much interest on this filed in Japan.
The head investigator proposed a new method for solving multi-dimensional inverse problems. The essential idea consists in embedding the inverse boundary value problem in R^n to the hyperbolic space. This new method introduced new view points of inverse problems. He discussed the inverse problem for the local perturbation of conformal metrics on hyperbolic manifolds. As an interestring by-product, he proved that in the boundary value problems in R^3, the electric conductivities can be identified locally from the knowledge of local Dirichlet-Neumann map. This hyperbolc space approach can also be applied to the problem of identification of locations of inclusions, and the reconstruction problem for linearized equations. They are expected to have applications to medical science. Okada studied numerical harmonic analysis, in particular, spline functions, wavelet analysis and numerical computation. Mochizuki studied inverse problems of reconstructing coefficients of Dirac operators from the data in finite intervals. Yoshitomi studied the properties of eigenvalues of the Laplacian on the 2-dimensional domain with crack, and also those of band region. Nakamaura studied the inverse problem of identifying the obstacle from the refelcted waves and also the inverse problem for elastic equations. Tamura studied the Aharonov-Bohm effect for the 2-diemnsional Schrodinger operators with Dirac type magnetic fields.
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