Mathematical analysis of scattering phenomena and inverse problems

本研究项目旨在发展由散射现象的数学分析引起的逆问题的研究。主要的主题是研究微分算子的谱。作为一个主要的会议项目，我们于2002年10月在京都组织了一个反问题国际研讨会，该领域的主要研究人员聚集在一起，并在日本促进了对该领域的兴趣。首席研究员提出了一种求解多维逆问题的新方法。其基本思想在于将R^n的反边值问题嵌入到双曲空间中。该方法引入了反问题的新观点。他讨论了双曲流形上共形度量的局部摄动逆问题。作为一个有趣的副产物，他证明了在R^3的边值问题中，电导率可以通过局部狄利克雷-诺伊曼映射的知识来局部识别。这种双曲空间方法也可以应用于夹杂物位置的识别问题，以及线性化方程的重构问题。它们有望应用于医学科学。冈田研究了数值调和分析，特别是样条函数、小波分析和数值计算。Mochizuki研究了用有限区间数据重构Dirac算子系数的逆问题。Yoshitomi研究了二维裂纹域上拉普拉斯算子的特征值性质，以及带域的特征值性质。Nakamaura研究了从反射波中识别障碍物的反问题和弹性方程的反问题。Tamura研究了二维薛定谔算子在狄拉克型磁场下的Aharonov-Bohm效应。

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.