Mathematical Sciences: Meromorphic Solutions of DifferentialEquations and Spectral Theory

该奖项支持对微分方程理论问题的数学研究，特别是与二阶方程的谱理论有关的研究。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，人们把重点放在了薛定谔算子谱的研究上。在20世纪60年代发现了有限间隙势的关系、weerstrass椭圆函数和逆问题之后，取得了根本性的进展。当证明了当且仅当满足Korteweg-deVriess层次结构中的某些方程时，势如何具有有限数量的带时，对有限间隙势的重新兴趣得到了加强。直到最近，所有的工作都集中在实值势上。20世纪80年代研究了由复值初始数据引起的等谱问题和Korteweg-deVriess流，标志着这一研究的起点。基于经典皮卡德定理，给出了求解亚纯椭圆型系数微分方程的新方法。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，它还揭示了在微分算子的代数和泛函解析意义上的谱性质与相关微分方程解的全局解析性质之间的意想不到的关系。微分方程是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，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***