Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

数学科学：无限维哈密顿系统

• 批准号: 9703847
• 负责人: Thomas Kappeler
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703847&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hamiltonian; Systems; Infinite

9703847 T. Kappeler 1. Hamiltonian systems of infinite dimension: The analysis of the symplectic structure of the phase space of completely integrable Hamiltonian systems of (in)finite dimension is very useful in the study of Hamiltonian perturbations. The investigator plans to apply prior work on action-angle variables for the Korteweg-deVries equation (KdV) to contribute to the investigation of the problem of the existence of analytic, quasiperiodic (in time) solutions of Hamiltonian perturbations of KdV or of any of the equations in the KdV hierarchy (KAM type theorem) and to study longtime stability of solutions of perturbed KdV equations (Nekhoroshev-type estimates). Further, the investigator plans to prove, in a generic situation, local existence of (generalized) action-angle variables for a large class of completely integrable Hamiltonian systems of infinite dimension. 2. Regularized determinants of elliptic operators: They appear in modern physics (functional integrals) as well as in geometry and topology (torsion, eta-invariant). The investigator plans to continue his work on techniques for analyzing regularized determinants and their applications to geometry and topology: Torsions for bordism and relative torsion (in the usual and the L2-setting); numerical computations of regularized determinants via a deformation; manifolds of determinant class and representation of the L2-torsion of a closed manifold M as a limit of a sequence of appropriately normalized torsions associated to a sequence of finite covers of M. 3. Smoothing of dispersive waves: Dispersive waves appear in many instances, e.g. in the study of water waves. The investigator plans to analyze microlocal smoothing properties for a large class of linear and nonlinear systems of dispersive evolution equations of Schroedinger type. In many different applications, such as propagation of signals along optical fibers (telecommunication), analysis of water waves and currents, and theoretical physics (celestial mechanics, quantum field theory), the basic underlying ideal models turn out to be integrable systems of finite or infinite dimension. To be useful for applications, perturbations of these ideal models have to be studied. Whereas integrable systems of finite dimension and their perturbations are relatively well understood, much remains to be done in the infinite dimensional case.

9703847 T. Kappeler 1. 无限维哈密顿系统：（中）有限维完全可积哈密顿系统相空间的辛结构分析在哈密顿扰动的研究中非常有用。 研究人员计划将先前关于 Korteweg-deVries 方程 (KdV) 的作用角变量的工作应用于 KdV 或任何哈密顿扰动的解析、准周期（时间）解的存在性问题的研究KdV 层次中的方程（KAM 型定理）并研究扰动 KdV 方程解的长期稳定性（Nekhoroshev 型估计）。 此外，研究者计划在一般情况下证明一大类无限维完全可积哈密顿系统的（广义）作用角变量的局部存在性。 2. 椭圆算子的正则行列式：它们出现在现代物理学（函数积分）以及几何和拓扑（扭转、eta 不变量）中。研究人员计划继续研究分析正则行列式的技术及其在几何和拓扑中的应用：棱镜扭转和相对扭转（在通常和 L2 设置中）；通过变形对正则化行列式进行数值计算；行列式类的流形和闭流形 M 的 L2 扭转的表示，作为与 M 的有限覆盖序列相关的适当归一化扭转序列的极限。 3. 色散波的平滑：色散波出现在许多情况下，例如在水波的研究中。 研究人员计划分析一大类薛定谔型色散演化方程的线性和非线性系统的微局域平滑特性。 在许多不同的应用中，例如信号沿光纤传播（电信）、水波和水流分析以及理论物理学（天体力学、量子场论），基本的理想模型是有限或有限的可积系统。无限维度。为了对应用有用，必须研究这些理想模型的扰动。尽管有限维可积系统及其扰动相对较好地被理解，但在无限维情况下仍有许多工作要做。