Nonselfadjoint Inverse Problems

非自伴随反问题

• 批准号: 0304280
• 负责人: Rudi Weikard
• 金额: $ 11.86万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0304280&HistoricalAwards=false
• 关键词: Nonselfadjoint; Inverse; Problems

Direct and inverse spectral and scattering problems of selfadjoint Sturm-Liouville operators are among the most studied subjects in mathematics. The status of nonselfadjoint Sturm-Liouville operators is by far not as complete even though they are currently under intensive investigation by a number of people. The present research project aspires to make a contribution in this area, in particular with regard to inverse problems. The main tool which sets the treatment of selfadjoint problems apart from others is the spectral theorem. The dire consequences of its absence can sometimes be overcome when it is assumed that the di.erential equation has (or its solutions have) certain structural properties. For example, Floquet theory guarantees a certain structure for the solutions of periodic equations, which in turn allows to draw conclusions for the spectrum which are very similar to the selfadjoint case (intervals become analytic arcs). Another class of such potentials are the so called algebro-geometric potentials which have been intensively investigated in the past few decades by many people including the PI. It is planned to apply the expertise gathered to obtain results for this kind of potentials and certain perturbations of them. In particular, recovery of the potential of a Schrodinger equation from the location of eigenvalues and resonances will be investigated.Physical laws are encoded by differential equations. The problem of obtaining solutions (or at least some of their properties) knowing the coefficients of the differential equation is usually called a direct problem. The inverse problem, on the other hand, is the problem of obtaining the coefficients from a certain knowledge about the solutions (often knowledge about spectral properties). The goal of the project is to investigate certain aspects of such problems. The differential equations investigated have widespread applications in physics and engineering, e.g. recovering material properties inside an object from measurements on the outside of the object. The solution of inverse problems is at the heart of medical and industrial imaging, mineral exploration, and earth quake studies to name a few.

自伴 Sturm-Liouville 算子的正谱和逆谱及散射问题是数学中研究最多的课题之一。尽管目前许多人正在对非自伴 Sturm-Liouville 算子进行深入调查，但它们的状况迄今为止还不完整。目前的研究项目希望在这一领域做出贡献，特别是在反问题方面。将自伴问题的处理与其他问题区分开来的主要工具是谱定理。当假设微分方程具有（或其解具有）某些结构特性时，有时可以克服其缺失的可怕后果。例如，Floquet 理论保证了周期方程解的特定结构，这反过来又允许对谱得出与自共轭情况非常相似的结论（区间变成解析弧）。另一类这样的势是所谓的代数几何势，在过去的几十年里，包括 PI 在内的许多人对它进行了深入研究。计划应用所收集的专业知识来获得此类潜力及其某些扰动的结果。特别是，将研究从特征值和共振的位置恢复薛定谔方程的势。物理定律由微分方程编码。在已知微分方程系数的情况下获得解（或至少其某些属性）的问题通常称为直接问题。另一方面，反问题是从有关解的特定知识（通常是有关光谱特性的知识）获得系数的问题。该项目的目标是调查此类问题的某些方面。研究的微分方程在物理和工程中具有广泛的应用，例如从物体外部的测量中恢复物体内部的材料属性。逆问题的解决是医学和工业成像、矿产勘探和地震研究等领域的核心。