Meromorphic functions and their applications

提出者打算用函数几何理论和拓扑学的方法继续他们对亚纯函数的研究，这种方法在他们以前的研究中已经带来了重要的结果。这项提案中的所有问题都是由NSF资助的提倡者先前的研究结果自然产生的。作者打算集中讨论以下具体问题：具有多项式和有理势的一维薛定谔算子的实谱轨迹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性；全纯动力学和薛定谔算子谱理论中某些亚纯函数的一般性质；区间系上不连续函数的多项式逼近。提出的研究将扩大我们对分析和数学物理中出现的超越方程的解的理解。亚纯函数是数学中使用的最基本的函数类，它的大多数应用。除了指数、余弦和正切等初等函数外，这类函数还包括物理和工程中不可缺少的更高的超越函数。作者以前在这个主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中得到了重要的应用。这一提议包含了由量子力学驱动的函数论的几个问题，提出者希望他们的结果能够加深对这一基本物理理论的理解。PI和COPI还将与研究生合作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.