Mathematical Sciences: Multidimensional Inverse Scattering Problems

数学科学：多维逆散射问题

• 批准号: 9217627
• 负责人: Tuncay Aktosun
• 金额: $ 6.14万
• 依托单位: North Dakota State University Fargo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9217627&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Multidimensional; Inverse; Scattering

This project consists of an investigation of the inverse scattering problems for the multidimensional Schrodinger equations when the potential is independent of energy and when the potential is proportional to energy. In the former case, the Wiener-Hopf factorization and the generalized Muskhelishvili- Vekua technique will be used to investigate the characterization of the scattering data, the relationship between bound states and partial indices, the inversion from partial data, and the stability of the inversion. In the latter case, the inverse scattering problem will be formulated as a Riemann-Hilbert problem with a discontinuity of almost periodic type and its solution will be obtained through a generalized Wiener-Hopf factorization technique. The solution of the inverse scattering problem with energy- independent potentials is equivalent to determining molecular, atomic, and nuclear forces in terms of the scattering data obtained in collision experiments, and the determination of these forces is one of the fundamental problems in physics. The inverse scattering problem with energy-dependent potentials is equivalent to the determination of the properties of a medium using only the measurements made on the boundary; such a problem has many important applications in wave propagation, seismology, nondestructive testing, oil exploration, atmospheric profile inversion, and other areas.

该项目包括对电势独立于能量以及电势与能量成比例的多维施罗宾格方程的反向散射问题的研究。 在前一种情况下，Wiener-HOPF分解和广义的Muskhelishvili-Vekua技术将用于研究散射数据的表征，界面与部分指标之间的关系，部分数据的反转以及反转的稳定性。 在后一种情况下，反向散射问题将被提出为riemann-hilbert问题，几乎是周期性类型的不连续性，其解决方案将通过广义的Wiener-HOPF分解技术获得。 与能量独立电位相反的散射问题的解相当于根据碰撞实验中获得的散射数据来确定分子，原子和核力量，而确定这些力是物理学的基本问题之一。 具有能量依赖性电势的逆散射问题等效于仅使用边界上的测量值确定培养基的性质。这样的问题在波浪传播，地震学，非破坏性测试，石油勘探，大气剖面反转和其他领域中具有许多重要的应用。