Mathematical models for dispersive electromagnetic waves: an overview <s>  <s> I. Preliminaries.- II. Pick Functions.- III. Pick Matrices and Loewner Determinants.- IV. Fatou Theorems.- V. The Spectral Theorem.- VI. One-Dimensional Perturbations.- VII. Monotone Matrix Functions.- VIII. Sufficient Conditions.- IX. Loewner's Theorem.- X. Reproducing Kernels.- XI. Nagy-Koranyi Proof of Loewner's Theorem.- XII. The Cauchy Interpolation Problem.- XIII. Interpolation by Pick Functions.- XIV. The Interpolation of Monotone Matrix Functions.- XV. Almost Positive Matrices.- XVI. The Analytic Continuation of Bergman Kernels.- XVII. The Loewner-FitzGerald Theorem.- XVIII. Loewner's Differential Equation.- XIX. More Analytic Continuation.- Notes and Comment. <s> BIB001 </s> Mathematical models for dispersive electromagnetic waves: an overview <s>  <s> Using a sum rule, we derive new bounds on Herglotz functions that generalize those given in (Gustafson and Sj\"oberg 2010) and (Bernland, Luger and Gustafson 2011). These bounds apply to a wide class of linear passive systems such as electromagnetic passive materials. Among these bounds, we describe the optimal ones and also discuss their meaning in various physical situations like in the case of a transparency window, where we exhibit sharp bounds. Then, we apply these bounds in the context of broadband passive cloaking in the quasi-static regime to negatively answer the following challenging question: is it possible to construct a passive cloaking device that cloaks an object over a whole frequency band? Our rigorous approach, although limited to quasi-statics, gives quantitative limitations on the cloaking effect over a finite frequency range by providing inequalities on the polarizabilty tensor associated with the cloaking device. We emphasize that our results hold for a cloak or object of any geometrical shape. <s> BIB002 </s> Mathematical models for dispersive electromagnetic waves: an overview <s>  <s> Abstract This work addresses the question of the construction of stable perfectly matched layers (PMLs) for a cold plasma in the infinitely large background magnetic field. We demonstrate that the traditional, Berenger's perfectly matched layers are unstable when applied to this model, due to the presence of the backward propagating waves. To overcome this instability, we use a combination of two techniques presented in the article. First of all, we consider a simplified 2D model, which shares with the 3D case one of the difficulties for the PML treatment, namely, the presence of the backward propagating waves. Based on the fact that for a fixed frequency either forward or backward propagating waves are present, we stabilize the PMLs with the help of a frequency-dependent correction. An extra difficulty of the 3D model compared to the 2D case is the presence of both forward and backward waves for a fixed frequency. To overcome this problem we construct a system of equations that consists of two independent systems, which are equivalent to the original model. The first of the systems behaves like the 2D plasma model, and hence the PMLs are stabilized again with the help of the frequency-dependent correction. The second system resembles the Maxwell equations in vacuum, and hence the standard Berenger's PMLs are stable for it. The systems are solved inside the perfectly matched layer, and coupled to the original Maxwell equations, which are solved in a physical domain, on a discrete level through an artificial layer. The numerical experiments confirm the stability of the new technique. <s> BIB003 </s> Mathematical models for dispersive electromagnetic waves: an overview <s>  <s> This work deals with Perfectly Matched Layers (PMLs) in the context of dispersive media, and in particular for Negative Index Metamaterials (NIMs). We first present some properties of dispersive isotropic Maxwell equations that include NIMs. We then demonstrate numerically the inherent instabilities of the classical PMLs applied to NIMs. We propose and analyse the stability of very general PMLs for a large class of dispersive systems using a new change of variable. We give necessary criteria for the stability of such models. For dispersive isotropic Maxwell equations, this analysis is completed by giving necessary and sufficient conditions of stability. Finally, we propose new PMLs that satisfy these criteria and demonstrate numerically their efficiency. <s> BIB004
and more especially the large time behaviour of the solution of the evolution problem with a time harmonic source. In BIB004 , [3] , BIB003 we addressed the question of the construction and analysis of stable Perfectly Matched Layers (PML's) for dispersive Maxwell's equations, for the time domain numerical simulation purpose. Finally, in BIB002 , we address the question of broadband passive cloaking, in other words, whether it possible to construct an electromagnetic passive cloak that cloaks an object over a whole frequency band. We answer negatively to this question in the so-called quasistatic regime and provide quantitative limitations on the cloaking effect over a finite frequency range. When working on this subject we have encountered two main difficulties. The first one is the absence of a work that would provide a unified, rigorous presentation and analysis of the existing mathematical models, despite the fact that many related publications can be found in a broad range of fields including applied and theoretical physics [27] , [32] , electric circuit theory and pure and applied mathematics  ,  , BIB001 . The second difficulty lies in the fact that, because of the abundance of the specialized literature it is not clear which statements are proven and which are simply commonly admitted. Thus, in the present work, we would like to partially fill these gaps. Properly speaking, this article is not a research paper. It has to be considered more as a review paper in which we try to gather the results from the literature that we found the most useful for applied mathematicians, provide an original presentation of these results and propose some new ideas (which, to our knowledge, have not occurred in the existing literature). We tried to keep the presentation rigorous, even though sometimes, for the sake of readability, we sacrificed formalism. Most proofs are detailed and only use elementary tools (and those that do not are postponed to appendix). In this way, the article is selfcontained and accessible to readers (physicists, engineers) who are not mathematicians. We hope that it can be seen as a useful toolbox for any scientist starting to study the subject, especially for applied mathematicians and numerical analysts. We are happy to dedicate this work to Peter Monk, who has been a major contributor of the numerical analysis of Maxwell's equations  , on the occasion of his 60 th birthday. We conclude this introduction by a brief outline of the rest of the paper. In Section 2 we formulate properties of the electric permittivity and magnetic permeability, studying them from mathematics and physics based points of view. In particular, we concentrate on the mathematical description of the so-called passivity property (Section 2.2) and discuss the relationship between its physical and mathematical interpretation in Section 2.3. In Section 3, we address the case when the permittivity and permeability are rational fractions (or 'local' materials, the name will be explained later). In the time domain they give rise to the Maxwell's equations coupled with ODEs. The results of this section include: the mathematical characterization of local passive materials (Section 3.1), the equivalence of passivity and well-posedness for a class of models (Section 3.2), a characterization of forward and backward propagating waves based on the analysis of the dispersion relation (Section 3.3). Finally, Section 4 is dedicated to the extension of the analysis of Section 3 to general passive media. 2 Mathematical models for dispersive electromagnetic waves