Research on Functional Analysis and Mathematical theory of Feynman path integrals.

费曼路径积分泛函分析与数学理论研究。

• 批准号: 15540184
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540184/
• 关键词: Feynman path integrals; Oscillatory integrals; Schrodinger equation; Stationary phase; Selfajoint operator; Quantum mechanics; WKB-method; path integrals

1. Fujiwara tried to give mathematically rigorous treatment of Feynman path integrals. He proved an improved remainder estimate of stationary phase method for oscillatory integrals over a space of large dimension. And he make the discussion of Kumano-go's results on convergence of Feynman path integrals. He also proved a new formula for the second term of the semi-classical asymptotics of Feynman path integrals.2. Yajima got results on spectrum and scattering properties of Nelson model, which is a simplified model of non-relativistic QED. He also succeeded in proving that every solution of Schrodinger equation with potentials which grow of oder O(|x|^m), m > 2 at the infinity gains, at almost every t, differentiability of order 1/m compared with its initial value.3. Watanabe made research on solutions of PDE with dispersive type. In the joint with T. Suzuki and T. Kobayashi he also proved interface regularity of solutions to the system of Maxwell-Stokes equations. He also discussed.4. Shimomura discussed large time behaviour of solutions to non-linear Schrodinger equations. He succeeded in finding a quite a fine property of solutions which is not foreseen from mere growth order of nonlinear term. Shimomura also proved smoothing effects of time local solution to non-linear Schrodinger equation with electro-magnetic potential.

1. 藤原试图给出费曼路径积分的严格数学处理方法。他证明了大尺度空间上振荡积分的一种改进的定相法剩余估计。并对熊野吾关于费曼路径积分收敛性的结论进行了讨论。他还证明了费曼路径积分的半经典渐近第二项的一个新公式。Yajima得到了非相对论性QED的简化模型Nelson模型的光谱和散射特性的结果。他还成功地证明了薛定谔方程的每一个解，其势能增长为O阶（|x|^m），在无穷远处，几乎在每一个t处，与初始值相比，可微性都是1阶/m。Watanabe对分散型PDE溶液进行了研究。他还与铃木和小林共同证明了麦克斯韦-斯托克斯方程组解的界面正则性。他还讨论了……Shimomura讨论了非线性薛定谔方程解的大时间行为。他成功地发现了解的一个很好的性质，这是仅仅从非线性项的增长顺序无法预见的。Shimomura还证明了具有电磁势的非线性薛定谔方程的时间局部解的平滑效应。

项目成果

Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions

• DOI: 10.3934/dcds.2003.9.1571
• 发表时间: 2003-09
• 期刊: Discrete and Continuous Dynamical Systems
• 影响因子: 1.1
• 作者: A. Shimomura

Nonexistence of asymptotically free solutions for quadratic nonlinear Schrodinger equations in two space dimensions

二维空间二次非线性薛定谔方程不存在渐近自由解

• 发表时间: 2005
• 期刊: Differential Integral Equations 18
• 作者: Shimomura;A.
• 通讯作者: A.

Scattering theory for Zakharov equations in three space dimensions with large data,

大数据空间三维扎哈罗夫方程的散射理论，

• 发表时间: 2004
• 期刊: Commun.Contemp.Math. 6
• 作者: Shimomura;A.
• 通讯作者: A.

MODIFIED WAVE OPERATORS FOR NONLINEAR SCHR¨ ODINGER EQUATIONS IN ONE AND TWO DIMENSIONS

• 发表时间: 2004
• 作者: N. Hayashi;P. Naumkin;A. Shimomura;S. Tonegawa

Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions

• DOI: 10.1619/fesi.47.63
• 发表时间: 2004
• 作者: A. Shimomura