Mathematical Sciences: Meromorphic Solutions of DifferentialEquations and Spectral Theory

Weikard 9401816 该奖项支持微分方程理论问题的数学研究，特别是与二阶方程的谱理论有关的问题。 这个问题有一个长期的，杰出的，历史上的重大重点放在研究的频谱薛定谔运营商在本世纪。 在20世纪60年代发现了有限间隙势、维尔斯特拉斯椭圆函数和反问题之间的关系之后，取得了根本性的进展。 当人们发现势可以有有限的能带数当且仅当它们满足Korteweg-deVriess方程时，人们对有限能隙势的兴趣又重新增强了。 直到最近，所有的工作都集中在实值势上。 等谱问题和由复值初值引起的Korteweg-deVriess流的研究始于20世纪80年代，标志着这一研究的开始。 基于Picard关于椭圆系数微分方程的一个经典定理，提出了一种新的研究方法，其解都是亚纯的。 这种方法有望成为一个强大的工具，用于解决该领域的一些开放的问题，如实际计算的带边（通过减少到线性代数特征值问题）和分类的椭圆有限间隙潜力的等谱流形。 此外，它揭示了一个意想不到的关系谱特性之间的代数以及在功能分析意义上的微分算子和全球分析性质的解决方案的相关微分方程。 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建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

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. ***