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 该奖项支持微分方程理论问题的数学研究，特别是与二阶方程谱理论相关的问题。 该学科有着悠久而杰出的历史，在本世纪的大部分时间里，薛定谔算子谱的研究受到了极大的重视。 继 20 世纪 60 年代发现有限间隙势、Weierstrass 椭圆函数和反演问题的关系之后，取得了根本性进展。 当证明势能具有有限数量的能带当且仅当它们满足 Korteweg-deVriess 层次结构中的某些方程时，人们对有限带隙势的新兴趣得到了增强。 直到最近，所有工作都集中在实际价值潜力上。 由复值初始数据产生的等谱问题和 Korteweg-deVriess 流在 20 世纪 80 年代被研究，标志着这项研究的起点。 基于皮卡德经典定理，我们提出了一种新方法，该定理涉及具有椭圆系数的微分方程，其所有解都是亚纯的。 这种方法有望成为解决该领域一些开放问题的有力工具，例如带边的实际计算（通过简化为线性代数特征值问题）和椭圆有限间隙势的等谱流形的分类。 此外，它揭示了代数以及微分算子的泛函分析意义上的谱特性与相关微分方程解的全局解析特性之间的意想不到的关系。 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。 ***