Mathematical Sciences: Meromorphic Solutions of DifferentialEquations and Spectral Theory

威卡德9401816该奖项支持对微分方程式理论问题的数学研究，特别是与二阶方程的谱理论有关的问题。这门学科有着悠久而杰出的历史，在本世纪的大部分时间里，人们都把重点放在了对薛定谔算子谱的研究上。S在60年代发现了有限间隔势、魏尔斯特拉斯椭圆函数和反问题之间的关系，取得了根本性的进展。当证明势能如何具有有限数目的能带当且仅当它们满足Korteweg-deVriess族中的某个方程时，人们对有限能隙势重新产生了兴趣。直到最近，所有的工作都集中在实值势能上。复值初值数据的等谱问题和Korteweg-deVriess流的研究始于20世纪80年代的S，标志着本文研究的起点。基于Picard关于椭圆型系数微分方程解均为亚纯的经典定理，提出了一种新的求解方法。这种方法有望成为解决该领域中一些公开问题的有力工具，例如带边的实际计算(通过归结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，它还揭示了微分算子的代数谱性质以及泛函分析意义下的谱性质与相关微分方程解的整体解析性质之间的意外关系。微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

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. ***