Mathematical Sciences: Meromorphic Solutions of DifferentialEquations and Spectral Theory

数学科学：微分方程的亚纯解和谱理论

• 批准号: 9401816
• 负责人: Rudi Weikard
• 金额: $ 6万
• 依托单位: University of Alabama at Birmingham
• 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: 1994-04-01 至 1997-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401816&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Meromorphic; Solutions; DifferentialEquations

Weikard 9401816 This award supports mathematical research on problems in the theory of differential equations, especially relating to the spectral theory of second order equations. The subject has a long, distinguished, history in which significant emphasis was placed on the study of the spectrum of Schrodinger operators during much of the current century. Fundamental advances were made following discoveries in the 1960's of the relationship of finite gap potentials, Weierstrass elliptic functions and inverse problems. The renewed interest in finite gap potentials was intensified when it was shown how potentials can have a finite number of bands if and only if they satisfy some equation in the Korteweg-deVriess hierarchy. Until very recently, all work focused on real-valued potentials. The isospectral problem and Korteweg-deVriess flow arising from complex-valued initial data was studied in the 1980's, marking the starting point for this research. A new approach has been opened based on a classical theorem of Picard concerning differential equations with elliptic coefficients all of whose solutions are meromorphic. This approach promises to be a powerful tool for tackling some of the open problems in the field, such as practical computation of band edges (by reduction to a linear algebraic eigenvalue problem) and the classification of isospectral manifolds of elliptic finite- gap potentials. Moreover it sheds light on an unexpected relationship between spectral properties in the algebraic as well as in the functional analytic sense of differential operators and global analytic properties of solutions of the associated differential equations. 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. ***

Weikard 9401816该奖项支持有关微分方程理论中问题的数学研究，尤其是与二阶方程的光谱理论有关。 该主题具有悠久，杰出的历史，在本世纪的大部分时间里，对施罗宾格运营商的频谱的研究都非常重视。 在1960年代发现有限差距电位，Weierstrass椭圆功能和反问题的关系之后，取得了根本的进步。 当且仅当它们满足Korteweg-Devriess层次结构中的某些方程时，对有限间隙电位的新兴趣就会加剧。 直到最近，所有工作都集中在实现的潜力上。 在1980年代研究了由复杂评估的初始数据引起的同一问题和korteweg-devriess流，这标志着这项研究的起点。 基于PICARD的经典定理，就具有椭圆系数的微分方程进行了一种新方法，所有方法均为其溶液是meromormormormormormormormorphic。 这种方法有望成为解决该领域的一些开放问题的强大工具，例如对频段边缘的实际计算（通过还原为线性代数特征值问题）和椭圆形有限距离潜力的同一歧管的分类。 此外，它阐明了代数中的光谱特性以及差分运算符的功能分析意义与相关微分方程解决方案的全局分析特性之间的意外关系。 微分方程构成了物理世界数学建模的基础。 数学分析的作用并不是创建方程，而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，光滑和成长的问题的答案。 此外，分析通常会开发出解决这些近似值准确性的解决方案和估计值的方法。 ***