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Research in Functional Analsys and Mathematical theory of Feynman path integrals.

费曼路径积分的泛函分析和数学理论研究。

基本信息

批准号：
11640180
负责人：
FUJIWARA Daisuke
金额：
$ 2.24万
依托单位：
Gakushuin University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

1. Fujiwara tried to give mathematically rigorous treatment of Feynman path integrals. It may seem possible to get a new proof of Kumanogo-Taniguchi theorem for actions with electro-magnetic fields if we cleverly conbine N.Kumanogo's method to that of our work which were published in 1997. Fujiwara wrote a textbook published by Springer Verlarg Tokyo in 1999. The english translation of the title is "Mathematical Method of Feynman path integrals".2. S.T.Kuroda together with P.Kurasov of Stockholm University showed that the Krein's formula describing relations of self-adjoint extension of two operators can be understood as resolvent equation between two operators. And they dicussed H_∈-perturbation theory.3. Mizutani studied finite element methods for parabolic nonlinear partial differential equations.4. Watanabe togeher with Kurasov of Stockholm Univ. studied H_4 realization of selfadjoint extension of operators.5. Sugano together with Kurata of Metropolitan Univ. studied fundamental solutions of uniformly elliptic operators with potentials in a class of functions which is a generalization of the class of positive polynomials. They suceeded in proving their fundamental solutions show good behviour in the weighted L^p space and Morrey classes. Using these facts, they proved a good estimates of distribution of eigen values and sharp information for order of decay of eigen functions of their elliptic operators.

1.藤原试图对费曼路径积分进行严格的数学处理。如果我们巧妙地将N.Kumanogo的方法与我们1997年发表的工作相结合，似乎有可能得到关于电磁场作用的Kumanogo-Taniguchi定理的新证明。藤原撰写了一本教科书，由Springer Verlarg东京公司于1999年出版。标题的英译名为《费曼路径积分的数学方法》。S.T.Kuroda和斯德哥尔摩大学的P.Kurasov证明了描述两个算子的自伴扩张关系的Krein公式可以理解为两个算子之间的预解方程。并讨论了H_∈微扰理论。水谷研究了抛物型非线性偏微分方程解的有限元方法。渡边与斯德哥尔摩大学的Kurasov一起。研究了算子自伴扩张的H_4实现。Sugano和大都会大学的仓田。研究了具有位势的一致椭圆算子在一类函数中的基本解，这类函数是正多项式类的推广。他们成功地证明了他们的基本解在加权的L、p空间和Morrey类中表现出良好的性能。利用这些事实，证明了椭圆算子本征值分布的一个很好的估计和本征函数衰减阶的清晰信息。