Mathematical Sciences: Inverse Spectral Problems and Meromorphic Solutions of Differential Equations

摘要提案：DMS- 9623121 PI：Gesztesy Gesztesy提出研究两个相关领域的微分方程有关的反谱理论和亚纯解的系统常微分方程有关的可积发展 方程第一个领域是连接到逆谱理论和表征的薛定谔方程和可积系统，如Korteweg-de弗里斯（KdV）方程的解决方案的潜在系数的isoprosper集。建立在最近开发的迹公式（多维）薛定谔运营商在适当的克莱因光谱位移函数和一般设备建设isospectral潜在的系数，他打算继续最近的工作反频谱问题的限制潜力和反频谱理论的短程相互作用。在第二个领域，他提出了一个战略，以解决问题的特点所有椭圆代数几何有限间隙解决方案的一般矩阵族完全可积的发展方程。特别是，Gesztesy建议继续工作，最近导致了解决这一问题的表征KdV层次的基础上，一个新开发的方法为中心的定理皮卡德。 逆谱理论在原子、分子和核物理学以及计算机断层扫描、声学、电磁学、介质的结构/材料性质的分辨率和航空技术等广泛领域中具有重要的应用。同样，完全可积方程在非线性光学中，特别是在基于孤子的光通信系统中起着至关重要的作用。Gesztesy的工作预计将对这些领域产生影响。

ABSTRACT Proposal: DMS- 9623121 PI: Gesztesy Gesztesy proposes to study two related areas in differential equations pertaining to inverse spectral theory and meromorphic solutions of systems of ordinary differential equations related to integrable evolution equations. The first area is connected to inverse spectral theory and the characterization of isospectral sets of potential coefficients for the Schrodinger equation and representations of solutions of integrable systems such as the Korteweg-de Vries (KdV) equation. Building upon recently developed trace formulas for (multi-dimensional) Schrodinger operators in terms of appropriate Krein spectral shift functions and a general device for constructing isospectral potential coefficients, he intends to continue recent work on the inverse spectral problem for confining potentials and inverse spectral theory for short-range interactions. In connection with the second area, he proposes a strategy to solve the problem of characterizing all elliptic algebro-geometric finite-gap solutions of general matrix hierarchies of completely integrable evolution equations. In particular, Gesztesy proposes to continue work which recently led to a solution of this characterization problem for the KdV hierarchy on the basis of a newly developed approach centered around a theorem of Picard. Inverse spectral theory has significant applications in atomic, molecular, and nuclear physics and in areas as wide-ranging as computer tomography, acoustics, electromagnetics, resolution of structural/material properties of media, and aviation technology. Similarly, completely integrable equations play a vital role in nonlinear optics and especially in soliton-based optical communications systems. Gesztesy' work can be expected to have an impact on these areas.