Transmission Eigenvalues and Inverse Scattering Theory

透射本征值问题是一个非自伴本征值问题，最初出现在逆散射理论中，但最近也出现在热声成像中。透射本征值在逆散射理论中的重要性在于，这些本征值可以从测量的散射数据确定，并且它们携带关于散射体的材料特性的信息。本建议涉及电磁散射理论中传输本征值的数学研究，以及这些结果在无损检测问题中的应用。 到目前为止，对透射本征值及其应用的研究主要集中在散射体为电介质的情况下。即使在这种有限的情况下，当散射体有空腔、裂纹或夹杂物时，也只能得到部分结果（这种情况在无损检测中当然很重要）。研究人员计划将这些现有的结果扩展到散射体可能具有空腔、裂纹或夹杂物以及吸收或色散的情况。这导致，除其他问题外，研究复杂的传输本征值和调查是否存在真实的本征值在这种情况下。 从可靠性和节约成本的角度来看，国防和制造领域的材料无损检测已成为一个越来越重要的领域。不幸的是，在许多具有国家重要性的地区，测试材料结构缺陷的简单有效的方法仍处于起步阶段。 在这项提议中，研究人员将从数学上研究使用新发现的数据集的可能性，该数据集可以通过电磁波对材料进行询问来获得。该数据集由所谓的“传输特征值”组成，这些特征值携带有关被测材料结构稳定性的信息。 为了利用这些本征值对实际材料进行无损检测，有必要检查被测材料部分导电的情况，即可以从询问电磁波中吸收能量，以及材料特性与频率相关的情况，即材料是“色散的”。 本项目致力于理解以下问题：1）关于可能具有缺陷的吸收和色散材料，透射本征值提供了什么信息; 2）如何从测量的散射数据有效地确定这些透射本征值。

The transmission eigenvalue problem is a non-selfadjoint eigenvalue problem that originally arose in inverse scattering theory but has also recently appeared in thermo-acoustic imaging. The importance of transmission eigenvalues in inverse scattering theory is that these eigenvalues can be determined from the measured scattering data and they carry information about the material properties of the scatterer. This proposal is concerned with a mathematical investigation of transmission eigenvalues in electromagnetic scattering theory and the application of these results to problems in nondestructive testing. Until now research on transmission eigenvalues and their applications has concentrated on the case when the scatterer is a dielectric. Even in this restricted case, only partial results are available when the the scatterer has cavities, cracks or inclusions (such situations are of course important in nondestructive testing). The investigators plan to extend these existing results for dielectrics to the case when the scatterer may have cavities, cracks or inclusions as well as being absorbing or dispersive. This leads, among other problems, to a study of complex transmission eigenvalues and an investigation of whether or not real eigenvalues exist in this case. The nondestructive testing of materials in the areas of defense and manufacturing has become an increasingly important area from the points of view of both reliability and cost saving. Unfortunately, in many areas of national importance simple and effective methods for testing material for structural imperfections is still in its infancy. In this proposal the investigators will mathematically examine the possibility of using a newly discovered data set that can be obtained from the interrogation of materials by electromagnetic waves. This data set consists of what are called "transmission eigenvalues" and such eigenvalues carry information about the structural stability of the material being tested. In order to utilize these eigenvalues for the nondestructive testing of realistic materials it is necessary to examine the case when the material being tested is partially conducting, i.e. can absorb energy from the interrogating electromagnetic wave as well as the case when the material properties are frequency dependent, i.e. the material is "dispersive". This project is devoted to the problem of understanding 1) what information transmission eigenvalues provide about absorbing and dispersive material which may have imperfections and 2) how these transmission eigenvalues can be effectively determined from the measured scattering data.