Transmission eigenvalues
Transmission eigenvalues
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DOI：
10.1088/0266-5611/29/10/100201
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发表时间：
2013
期刊：
影响因子：
2.1
通讯作者：
F. Cakoni;H. Haddar
中科院分区：
文献类型：
--
作者：
F. Cakoni;H. Haddar
在逆散射理论中，传输特征值可以被视为不可穿透物体的谐振频率概念到可穿透电介质情况的延伸。传输特征值问题是偏微分方程谱理论的一个相对较晚出现的问题。它的首次出现是在 1986 年 Kirsch 的一篇论文中，当时 Kirsch 正在研究亥姆霍兹方程散射解的远场图案的密集性，或者用更现代的术语来说，远场算子的单射性 [1]。 Kirsch 的论文很快被 Colton 和 Monk 进行了更系统的研究，他们开发了双空间方法来解决非均匀介质中声波的逆散射问题 [2]。在本文中，他们表明对于球形分层媒体传输特征值存在并形成离散集。还给出了数值示例，表明原则上可以根据远场数据确定传输特征值。 Colton 等人于 1989 年 [3] 以及 Rynne 和 Sleeman 于 1991 年 [4] 发表的论文总结了对传输特征值的第一个兴趣阶段，该论文表明，对于非均匀介质（不一定是球形分层），传输特征值（如果存在）会形成一个离散集。在接下来的十七年里，传输特征值被忽略。这主要是因为，随着各种采样方法的引入，从远场数据确定非均匀介质的形状，传输特征值是需要避免的，因此传输特征值最多形成一个离散集合被认为是足够的。此外，由于特征值问题的非线性和相关传输特征值问题的特殊结构，与传输特征值的存在或相关特征向量的结构相关的问题被认为是特别困难的。在 Cakoni 等人 [5] 和 Cakoni 等人 [6] 发表一系列论文后，回答这些问题的必要性变得很重要，这些论文表明这些传输特征值可用于从远场数据中获取有关散射物体材料特性的定性信息。一般情况下传输特征值存在性的第一个答案是在 2008 年给出的，当时 Päivärinta 和 Sylvester 证明了折射率足够大时存在传输特征值 [7]，随后 Cakoni 等人在 2010 年发表了论文，取消了折射率的大小限制 [8]。更重要的是，后者表明透射特征值产生了关于散射物体材料特性的定性信息，并且 Cakoni 等人在 [9] 中确立了透射特征值可以通过远场方程的 Tikhonov 正则化解来确定。自从这些论文出现以来，人们对传输特征值问题产生了浓厚的兴趣（我们建议读者参阅我们最近的调查论文 [10]，详细了解该领域截至 2012 年的发展），本期特刊中的论文代表了这项研究所采取的众多方向。事实上，我们很高兴看到[10]中提出的许多开放性理论和数值问题在本期特刊的贡献中得到了（全部或部分）回答：对比度最小假设下的透射本征值的存在、透射本征值的数值评估、逆谱问题、无损检测的应用等。除了这些主题之外，还提出了许多其他新的调查和研究方向，正如我们将在下面的简短内容摘要中看到的那样。本期特刊中的许多论文都涉及传输特征值的存在问题以及相关传输特征函数的结构。 Robbiano [11]、Blasten 和 Päivärinta [12] 以及 Lakshtanov 和 Vainberg [13] 的三篇论文在折射率（可能是复值）弱假设下提供了关于标量问题传输特征值存在性的新的补充结果，该假设主要规定对比度不会改变边界上的符号。有趣的是，这里看到三种不同的新方法来获得这些结果。另一方面，Bonnet-Ben Dhia 和 Chesnel [14] 的论文解决了当对比度在边界上改变符号时内部传输问题的 Fredholm 特性，展示了该特性失效的情况。 Delbary [15] 的论文使用更标准的方法，在麦克斯韦方程组的背景下分析了频率相关材料的传输特征值的存在和结构，而 Vesalainen [16] 的论文通过考虑具有非紧支持势的薛定谔方程的传输特征值，开始研究无界域中的传输特征值问题。 Monk 和 Selgas [17] 的论文讨论了电介质安装在完美导体上的情况，并提供了一些使用线性采样方法定位相关特征值的数值示例。随后，一系列论文解决了球形分层介质的传输特征值局部化问题以及相关的逆谱问题。更具体地说，Colton 和 Leung [18] 的论文提供了关于复传输特征值的新结果，并为逆谱问题解的唯一性提供了新的证明，而 Sylvester [19] 的论文提供了关于如何在折射率恒定时定位与角度无关本征函数相关的所有传输特征值的清晰结果。 Gintides 和 Pallikarakis [20] 的论文研究了一种迭代最小二乘法，用于从透射特征值中识别球面分层折射率。在远场测量方面的传输特征值表征方面，Kirsch 和 Lechleiter [21] 获得了一个有希望的新结果，展示了如何使用可根据测量的散射数据获得的散射算子的特征值来识别传输特征值。在 Kleefeld [22] 的论文中，提出了一种基于内部传输问题的表面积分公式和非线性特征值问题的数值方法计算传输特征值的精确方法，并针对三维标量问题进行了数值验证。另一方面，Sun 和 Xu 的论文 [23] 研究了使用与内部传输问题的变分公式相关的标准迭代方法来计算麦克斯韦方程组的传输特征值，重点是各向异性对传输特征值的影响。 Cakoni和Moskow的论文[24]从在无损检测中使用传输特征值的角度出发，研究了传输特征值对于小的不均匀性的渐近行为。 Nakamura 和 Wang [25] 的论文研究了时间相关热方程的线性采样方法，并分析了与该方程相关的内部传输问题。最后，在 Finch 和 Hickmann [26] 的论文中，内部传输问题的频谱与热声成像中物体声学特性的独特确定有关。我们希望这本论文集能够促进快速发展的传输特征值和逆散射理论领域的进一步研究。
In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission eigenvalue problem. The need to answer these questions became important after a series of papers by Cakoni et al [5], and Cakoni et al [6] suggesting that these transmission eigenvalues could be used to obtain qualitative information about the material properties of the scattering object from far-field data. The first answer to the existence of transmission eigenvalues in the general case was given in 2008 when Päivärinta and Sylvester showed the existence of transmission eigenvalues for the index of refraction sufficiently large [7] followed in 2010 by the paper of Cakoni et al who removed the size restriction on the index of refraction [8]. More importantly, in the latter it was shown that transmission eigenvalues yielded qualitative information on the material properties of the scattering object and Cakoni et al established in [9] that transmission eigenvalues could be determined from the Tikhonov regularized solution of the far-field equation. Since the appearance of these papers there has been an explosion of interest in the transmission eigenvalue problem (we refer the reader to our recent survey paper [10] for a detailed account of the developments in this field up to 2012) and the papers in this special issue are representative of the myriad directions that this research has taken. Indeed, we are happy to see that many open theoretical and numerical questions raised in [10] have been answered (totally or partially) in the contributions of this special issue: the existence of transmission eigenvalues with minimal assumptions on the contrast, the numerical evaluation of transmission eigenvalues, the inverse spectral problem, applications to non-destructive testing, etc. In addition to these topics, many other new investigations and research directions have been proposed as we shall see in the brief content summary below. A number of papers in this special issue are concerned with the question of existence of transmission eigenvalues and the structure of the associated transmission eigenfunctions. The three papers by respectively Robbiano [11], Blasten and Päivärinta [12], and Lakshtanov and Vainberg [13] provide new complementary results on the existence of transmission eigenvalues for the scalar problem under weak assumptions on the (possibly complex valued) refractive index that mainly stipulates that the contrast does not change sign on the boundary. It is interesting here to see three different new methods to obtain these results. On the other hand, the paper by Bonnet-Ben Dhia and Chesnel [14] addresses the Fredholm properties of the interior transmission problem when the contrast changes sign on the boundary, exhibiting cases where this property fails. Using more standard approaches, the existence and structure of transmission eigenvalues are analyzed in the paper by Delbary [15] for the case of frequency dependent materials in the context of Maxwell's equations, whereas the paper by Vesalainen [16] initiates the study of the transmission eigenvalue problem in unbounded domains by considering the transmission eigenvalues for Schrödinger equation with non-compactly supported potential. The paper by Monk and Selgas [17] addresses the case where the dielectric is mounted on a perfect conductor and provides some numerical examples of the localization of associated eigenvalues using the linear sampling method. A series of papers then addresses the question of localization of transmission eigenvalues and the associated inverse spectral problem for spherically stratified media. More specifically, the paper by Colton and Leung [18] provides new results on complex transmission eigenvalues and a new proof for uniqueness of a solution to the inverse spectral problem, whereas the paper by Sylvester [19] provides sharp results on how to locate all the transmission eigenvalues associated with angular independent eigenfunctions when the index of refraction is constant. The paper by Gintides and Pallikarakis [20] investigates an iterative least square method to identify the spherically stratified index of refraction from transmission eigenvalues. On the characterization of transmission eigenvalues in terms of far-field measurements, a promising new result is obtained by Kirsch and Lechleiter [21] showing how one can identify the transmission eigenvalues using the eigenvalues of the scattering operator which are available in terms of measured scattering data. In the paper by Kleefeld [22], an accurate method for computing transmission eigenvalues based on a surface integral formulation of the interior transmission problem and numerical methods for nonlinear eigenvalue problems is proposed and numerically validated for the scalar problem in three dimensions. On the other hand, the paper by Sun and Xu [23] investigates the computation of transmission eigenvalues for Maxwell's equations using a standard iterative method associated with a variational formulation of the interior transmission problem with an emphasis on the effect of anisotropy on transmission eigenvalues. From the perspective of using transmission eigenvalues in non-destructive testing, the paper by Cakoni and Moskow [24] investigates the asymptotic behavior of transmission eigenvalues with respect to small inhomogeneities. The paper by Nakamura and Wang [25] investigates the linear sampling method for the time dependent heat equation and analyses the interior transmission problem associated with this equation. Finally, in the paper by Finch and Hickmann [26], the spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. We hope that this collection of papers will stimulate further research in the rapidly growing area of transmission eigenvalues and inverse scattering theory.