Mathematical Analysis of Quantum Physics

量子物理的数学分析

• 批准号: 14340039
• 负责人: YAJIMA Kenji
• 金额: $ 5.31万
• 依托单位: Gakushuin University (2003-2005)The University of Tokyo (2002)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340039/
• 关键词: Mathematical Physics; Partial differential equations; Nonlinear differential equations; Spectral theory; Scattering theory; Time periodic system; Microlocal singularities; Random system; 量子力学; シュレーディンガー方程式; 散乱の波動作用素; 特異性伝播; 経路積分; 非線形方程式; スペク

Various linear and non-linear partial differential equations and selfadjoint operators which appear in quantum physics have been investigated from wide range of spectrum by various methods. Principal results obtained in this period of research are the following1.The boundedness in Lebesgue spaces and Sobolev spaces of the wave operators of scattering theory for Schroedinger operators has been proved when spatial dimension is larger or equal to three including the case when the Schroedinger operator has the threshold singularities at the bottom of the continuous spectrum. The result is applied for the dispersive estimates for the solutions of the corresponding time dependent Schroedinmger equations2.For the solutions of time dependent Schroedinger equations the following results have been obtained. (1)The large time asymptotic formula is obtained for the first time in the case that the operator has the threshold singularities at the bottom of the continuous spectrum. (2)It is found and … More proved rigorously that the degree of the smoothing effect for the solutions is determined by the asymptotic behavior of the sojourn time in bounded sets of the corresponding classical particles when the energy becomes unbounded indefinitely. (3)For describing the propagation of micro-local singularities, the new notion of the homogeneous wave front sets is introduced and existing results on the propagation of analytic singularity or the singularities in the category of infinitely differentiability have been considerably improved, (4)It is found and proved rigorously that, in the semi-classical limit, solutions may be expanded in terms of resonances in the domains bounded by high wall of the potentials3.The extended phase method or the Floquet Hamiltonian method for studying time periodic system has been improved and, by using this, the asymptotic expansions of scattering solutions of time periodic Schroedinger equations have been obtained.4.The asymptotic completeness of scattering theory in the one photon sector of Nelson model, a simplified model of quantum electro-dynamics has been proved and the nature of the spectrum of the Hamiltonian in that sector has been determined.5.Mathematically rigorous definition of Feynman path integral has been given for the first time when the integrands are polynomially increasing functionals and the second term in the semi-classical expansion is obtained. Precise error estimates for the stationary phase method of oscillatory integrals in a large dimensional space have been obtained.6.The nature of the spectrum and the behavior of integrated density of states as the smoothness of Schroedinger operators with random potentials or random magnetic fields of specific types have been obtained. Less

本期研究的主要成果如下： 1、Lebesgue空间和Sobolev空间的有界性。证明了薛定谔算子在空间维度大于或等于3时的散射理论的波算子，包括薛定谔算子在连续谱底部具有阈值奇点的情况，并将结果应用于(1)在算子有(2)的情况下首次得到了大时间渐近公式 (2)发现并且……更严格地说，解的平滑效果的程度是由逗留时间(3)为了描述微局域奇点的传播，引入了齐次波前集的新概念以及解析奇点或奇点传播的现有结果。在无限可微性范畴内得到了相当大的改进，（4）严格地发现并证明了，在半经典极限下，解可以在以高壁为边界的域中以共振形式展开。 3.改进了研究时间周期系统的扩展相位法或Floquet哈密顿法，并利用它得到了时间周期薛定谔方程的散射解的渐近展开式。4.散射理论在时间周期系统中的渐近完备性纳尔逊模型的一个光子扇区，量子电动力学的简化模型已被证明，并且该扇区中哈密顿量的光谱性质已被确定。5.数学上严格的定义首次给出了被积函数为多项式增函数时的费曼路径积分，并得到了半经典展开式的第二项，得到了大维空间中振荡积分的平稳相位法的精确误差估计。 6.得到了特定类型的随机势或随机磁场下的能谱性质和积分态密度作为薛定谔算子的平滑性的行为。

项目成果

中村 周: "フーリエ解析(応用数学基礎講座4)"朝倉書店. 186 (2003)

中村秀：《傅里叶分析（基础应用数学课程4）》朝仓书店186（2003）。

T.Kobayashi: "Interface regularity for the Maxwell and Stokes systems"Osaka Journal of Mathematics. 40. 925-944 (2003)

T.Kobayashi：“麦克斯韦和斯托克斯系统的接口正则性”大阪数学杂志。

フーリエ解析(応用数学基礎講座4)

傅立叶分析（基础应用数学课程4）

• 发表时间: 2003
• 作者: Araki;H.;Ezawa;H.;中村 周
• 通讯作者: 中村 周

The Nelson model less than two photons

尼尔森模型少于两个光子

• 发表时间: 2003
• 期刊: Annales of l' Institues Henri Poincare 4
• 作者: Toshiyuki Kobayashi;Salma Nasrin;A.Galtbayer
• 通讯作者: A.Galtbayer

Phase space tunneling and multistate scattering..

相空间隧道和多态散射..

• 发表时间: 2002
• 期刊: Journal of Functional Analysis 191
• 作者: A.Shimomura;A.Shimomura;A.Shimomura;A Jensen;J-M.Combes;S.Nakamura;A.Martinez
• 通讯作者: A.Martinez