Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

数学科学：无限维哈密顿系统

• 批准号: 9401020
• 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: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401020&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hamiltonian; Systems; Infinite

9401020 Kappeler This award supports mathematical research focusing on infinite systems of differential equations known as Hamiltonian systems. They derive from models of physical systems and nonlinear partial differential equations obtained through variational principles. Work to be done includes the analysis of the symplectic structure of the phase space of completely integrable Hamiltonian systems to help understand Hamiltonian perturbations. Studies of regularized determinants of elliptic operators will also be carried out. They appear in modern physics (functional integrals) as well as in geometry and topology (torsion and eta-invariants). This work continues efforts to extend results on asymptotic expansions of the logarithm of the determinant of one-parameter families of pseudodifferential operators. A third line of investigation concerns the investigation of microlocal smoothing properties for a large class of linear and nonlinear systems of dispersive evolution equations of Schrodinger type. Finally, work will bedone on spectral problems and related inverse problems for Schrodinger operators on line bundles over tori and Heisenberg manifolds. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***

9401020 Kappeler 该奖项支持专注于微分方程无限系统（称为哈密顿系统）的数学研究。 它们源自物理系统模型和通过变分原理获得的非线性偏微分方程。 要做的工作包括分析完全可积哈密顿系统相空间的辛结构，以帮助理解哈密顿扰动。 椭圆算子正则化行列式的研究也将进行。 它们出现在现代物理学（泛函积分）以及几何学和拓扑学（扭转和 eta 不变量）中。 这项工作继续努力扩展伪微分算子的单参数族行列式的对数渐近展开的结果。 第三个研究方向涉及薛定谔型色散演化方程的一大类线性和非线性系统的微局域平滑特性的研究。 最后，我们将致力于解决环面流形和海森堡流形上的线束上的薛定谔算子的谱问题和相关反演问题。 偏微分方程构成了物理世界数学建模的基础。 数学分析的作用与其说是创建方程，不如说是提供有关解决方案的定性和定量信息。 这可能包括有关独特性、平滑性和成长性问题的答案。 此外，分析通常会开发近似解的方法并估计这些近似的准确性。 ***