Meromorphic functions and their applications

摘要：提出者打算继续他们的亚纯函数的研究，使用几何理论的功能和拓扑的方法，已经带来了显着的成果，在他们以前的研究方法。该提案中的所有问题都是由NSF资助的提案人先前研究的结果自然产生的。提出者打算集中讨论下列具体问题：具有多项式和有理位势的一维薛定谔算子的真实的和复谱轨迹的几何和拓扑;由整关系定义的隐解析函数的奇异性;在全纯动力学和薛定谔算子谱理论中出现的某些亚纯函数类的一般性质;区间系上不连续函数的多项式逼近亚纯函数是数学中最基本的一类函数，也是数学中最重要的一类函数，它的应用范围非常广泛。除了像指数、余弦和正切这样的初等函数外，这类函数还包括在物理学和工程学中不可或缺的高等超越函数。作者以前在这个问题上的工作已经在控制理论，材料科学，计算机科学，信号处理，数学物理和天体物理学中找到了重要的应用。这个提议包含了由量子力学激发的函数理论的几个问题，提议者希望他们的结果能对这个基本物理理论产生更深入的理解。PI和coPI还将与研究生一起进行与该项目相关的研究。

Abstract.The proposers intend to continue their study of meromorphic functions using the methods of geometric theory of functions and topology, the approach which already brought significant results in their previous research. All problems in this proposal arise naturally from the results of the previous research of the proposers funded by NSF. The proposers intend to concentrate on the following specific problems: geometry and topology of real and complex spectral loci of the one-dimensional Schrödinger operators with polynomial and rational potentials; singularities of implicit analytic functions defined by entire relations; general properties of certain classes of meromorphic functions occurring in holomorphic dynamics and in spectral theory of the Schrödinger operators; polynomial approximation of discontinuous functions on systems of intervals. Proposed research will expand our understanding of solutions of transcendental equations emerging in analysis and mathematical physics.Meromorphic functions constitute the most basic class of functions used in mathematics and most of its applications. In addition to elementary functions like the exponent, cosine and tangent, this class includes higher transcendental functions, indispensable in physics and engineering. Previous work of the authors on this subject already found important applications in control theory, material science, computer science, signal processing, mathematical physics and astrophysics. This proposal contains several problems of the function theory motivated by quantum mechanics, and the proposers expect their results to yield deeper understanding of this fundamental physical theory. The PI and coPI will also work with graduate students on research related to the project.