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

数学科学：反谱问题和微分方程的亚纯解

• 批准号: 9623121
• 负责人: Friedrich Gesztesy
• 金额: $ 4.19万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623121&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Spectral; Problems

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.

摘要建议：DMS-9623121 PI：Gesztesy Gesztesy建议研究微分方程组中与逆谱理论有关的两个相关领域，以及与可积发展方程有关的常微分方程组的亚纯解。第一个领域与逆谱理论和薛定谔方程等谱势系数集的刻画以及可积系统的解的表示有关，如KdV方程。基于最近发展的(多维)薛定谔算符的适当Krein谱位移函数的迹公式和构造等谱势系数的通用装置，他打算继续最近在禁闭势的逆谱问题和短程相互作用的逆谱理论方面的工作。针对第二个领域，他提出了一种策略来刻画完全可积发展方程的一般矩阵族的所有椭圆代数几何有限间隙解。具体地说，Gesztesy建议继续最近导致KdV族的这个表征问题的解决的工作，该工作基于以Picard的一个定理为中心的新发展的方法。逆谱理论在原子、分子和核物理以及计算机层析成像、声学、电磁学、介质结构/材料性质解析和航空技术等领域有着重要的应用。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。可以预期，格兹特西的工作将对这些领域产生影响。