Nonselfadjoint Inverse Problems

自伴Sturm-Liouville算子的正、逆谱和散射问题是数学中研究最多的课题之一。非自伴Sturm-Liouville算子的状态至今还不完全，尽管它们目前正受到许多人的深入研究。本研究项目希望在这一领域作出贡献，特别是在逆问题。将自伴问题的处理与其他问题区别开来的主要工具是谱定理。当假定微分方程具有（或其解具有）某些结构性质时，它的缺失所带来的可怕后果有时可以得到克服。例如，Floquet理论保证了周期方程解的某种结构，这反过来又允许得出与自伴情况非常相似的谱结论（区间成为解析弧）。另一类这样的潜力是所谓的代数几何潜力已深入研究，在过去的几十年里，许多人，包括PI。计划应用所收集的专门知识，以获得这种潜力及其某些扰动的结果。特别地，将研究从本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。已知微分方程的系数而获得解（或至少获得解的某些性质）的问题通常被称为正问题。另一方面，逆问题是从关于解的一定知识（通常是关于谱特性的知识）获得系数的问题。该项目的目标是调查这些问题的某些方面。所研究的微分方程在物理学和工程学中有着广泛的应用，例如，从物体外部的测量中恢复物体内部的材料特性。反问题的解决方案是医学和工业成像，矿产勘探和地震研究的核心。

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.