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This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in [Cassier, Joly, Rosas Mart\'inez, Z. Angew. Math. Phys. 74 (2023), 115], we develop a method based on the spectral analysis of the underlying non-self-adjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation., Comment: 68 pages, 4 figures

It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances., Comment: 33 pages, 1 figure

We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that allow to quantify the quality of the cloaking effect. We illustrate our results by applying them to achieve active exterior cloaking for the heat equation., Comment: 32 pages, 11 figures

This paper is concerned with the time-dependent Maxwell's equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to $-1$ across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem., Comment: 64 pages, 4 figures

We study the band structure of self-adjoint elliptic operators $\mathbb{A}_g= -\nabla \cdot \sigma_{g} \nabla$, where $\sigma_g$ has the symmetries of a honeycomb tiling of $\mathbb{R}^2$. We focus on the case where $\sigma_{g}$ is a real-valued scalar: $\sigma_{g}=1$ within identical, disjoint "inclusions", centered at vertices of a honeycomb lattice, and $\sigma_{g}=g \gg1 $ (high contrast) in the complement of the inclusion set (bulk). Such operators govern, e.g. transverse electric (TE) modes in photonic crystal media consisting of high dielectric constant inclusions (semi-conductor pillars) within a homogeneous lower contrast bulk (air), a configuration used in many physical studies. Our approach, which is based on monotonicity properties of the associated energy form, extends to a class of high contrast elliptic operators that model heterogeneous and anisotropic honeycomb media. Our results concern the global behavior of dispersion surfaces, and the existence of conical crossings (Dirac points) occurring in the lowest two energy bands as well as in bands arbitrarily high in the spectrum. Dirac points are the source of important phenomena in fundamental and applied physics, e.g. graphene and its artificial analogues, and topological insulators. The key hypotheses are the non-vanishing of the Dirac (Fermi) velocity $v_D(g)$, verified numerically, and a spectral isolation condition, verified analytically in many configurations. Asymptotic expansions, to any order in $g^{-1}$, of Dirac point eigenpairs and $v_D(g)$ are derived with error bounds. Our study illuminates differences between the high contrast behavior of $\mathbb{A}_g$ and the corresponding strong binding regime for Schroedinger operators., Comment: 63 pages, 13 figures

We present an active cloaking method for the parabolic heat (and mass or light diffusion) equation that can hide both objects and sources. By active we mean that it relies on designing monopole and dipole heat source distributions on the boundary of the region to be cloaked. The same technique can be used to make a source or an object look like a different one to an observer outside the cloaked region, from the perspective of thermal measurements. Our results assume a homogeneous isotropic bulk medium and require knowledge of the source to cloak or mimic, but are in most cases independent of the object to cloak., Comment: 29 pages,12 figures

The purpose of this paper is to analysis the Harnack part of some truncated shifts whose $\rho$-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the $\rho$-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given.

Photonic crystals are periodic structures which prevent light propagation along one or more directions in certain frequency intervals. Their band spectrum is usually analyzed using Floquet-Bloch decomposition. This spectrum is located on the real axis, and it enters the complex plane when absorption and dispersion is considered in the dielectric permittivity of material constituents. Here, we review fundamental definition and properties of dispersion law and group velocity in photonic crystals and we illustrate them with numerical examples., Comment: book chapter

We present a method for imaging small scatterers in a homogeneous medium from polarization measurements of the electric field at an array. The electric field comes from illuminating the scatterers with a point source with known location and polarization. We view this problem as a generalized phase retrieval problem with data being the coherency matrix or Stokes parameters of the electric field at the array. We introduce a simple preprocessing of the coherency matrix data that partially recovers the ideal data where all the components of the electric field are known for different source dipole moments. We prove that the images obtained using an electromagnetic version of Kirchhoff migration applied to the partial data are, for high frequencies, asymptotically identical to the images obtained from ideal data. We analyze the image resolution and show that polarizability tensor components in an appropriate basis can be recovered from the Kirchhoff images, which are tensor fields. A time domain interpretation of this imaging problem is provided and numerical experiments are used to illustrate the theory., Comment: 39 pages, 13 figures

We study the surface diffusion of the model cyanobacterium Synechocystis sp. PCC $6803$ during the incipient stages of cell contact with a glass surface in the dilute regime. We observe a twitching motility with alternating immobile "tumble" and mobile "run" periods, resulting in a normal diffusion described by a continuous time random walk with a coefficient of diffusion $D$. Surprisingly, $D$ is found to decrease with time down to a plateau. This is observed only when the cyanobacterial cells are able to produce released extracellular polysaccharides, as shown by a comparative study between the wild-type strain and various polysaccharides-depleted mutants. The analysis of the trajectories taken by the bacterial cells shows that the temporal characteristics of their intermittent motion depend on the instantaneous fraction of visited sites during diffusion. This describes quantitatively the time dependence of $D$, related to the progressive surface coverage by the polysaccharides. The observed slowdown of the surface diffusion may constitute a basic precursor mechanism for microcolony formation and provides clues for controlling biofilm formation., Comment: 9 pages, 9 figures, to be published in Physical Review E

Using Spitzer photometry at 4.5 microns, we search for rings and satellites around the long period transiting planet CoRoT-9b. We observed two transits in 2010 and 2011. From their non-detection, we derive upper limits on the plausible physical characteristics of these objects in the planet environment. We show that a satellite larger than about 2.5 Earth radii is excluded at 3-$\sigma$ for a wide range of elongations at the two epochs of observations. Combining the two observations, we conclude that rings are excluded for a wide variety of sizes and inclination. We find that for a ring extending up to the Roche limit, its inclination angle from the edge-on configuration as seen from the Earth must be lower than 13{\deg} in the case of silicate composition and lower than 3{\deg} in the case of material with water ice density., Comment: Accepted for publication in A&A

In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background., Comment: 46 pages, 16 figures

We present a method for imaging the polarization vector of an electric dipole distribution in a homogeneous medium from measurements of the electric field made at a passive array. We study an electromagnetic version of Kirchhoff imaging and prove, in the Fraunhofer asymptotic regime, that range and cross-range resolution estimates are identical to those in acoustics. Our asymptotic analysis provides error estimates for the cross-range dipole orientation reconstruction and shows that the range component of the dipole orientation is lost in this regime. A naive generalization of the Kirchhoff imaging function is afflicted by oscillatory artifacts in range, that we characterize and correct. We also consider the active imaging problem which consists in imaging both the position and polarizability tensors of small scatterers in the medium using an array of collocated sources and receivers. As in the passive array case, we provide resolution estimates that are consistent with the acoustic case and give error estimates for the cross-range entries of the polarizability tensor. Our theoretical results are illustrated by numerical experiments., Comment: 35 pages, 18 figures

The purpose of this paper is to describe the Harnack parts for the operators of class C $\rho$ ($\rho$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with $\rho$-numerical radius one. The case of operators with $\rho$-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact $\rho$-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.

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., Comment: 36 pages, 2 figures

We explore the spectral properties of the time-dependent Maxwell's equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. We construct explicitly a generalized Fourier transform which diagonalizes the Hamiltonian that describes the propagation of transverse electric waves. This transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem. It will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles., Comment: 34 pages, 8 figures

In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we give a rigorous derivation of the field equation recursion method in the abstract theory of composites to two-component composites with isotropic phases. This method is of great interest since it has proven to be a powerful tool in developing sharp bounds for the effective tensor of a composite material. The reason is that the effective tensor $\bf L_*$ can be interpreted in the general framework of the abstract theory of composites as the $Z$-operator on a certain orthogonal $Z(2)$ subspace collection. The base case of the recursion starts with an orthogonal $Z(2)$ subspace collection on a Hilbert space $\cal H$, the $Z$-problem, and the associated $Y$-problem. We provide some new conditions for the solvability of both the $Z$-problem and the associated $Y$-problem. We also give explicit representations of the associated $Z$-operator and $Y$-operator and study their analytical properties. An iteration method is then developed from a hierarchy of subspace collections and their associated operators which leads to a continued fraction representation of the initial effective tensor $\bf L_*$., Comment: 27 pages; this is a copy of chapter 10 in the book "Extending the Theory of Composites to Other Areas of Science" edited by Graeme W. Milton, 2016, Milton-Patton Publishers, ISBN: 978-1-4835-6919-2

In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we derive the analyticity properties of the electromagnetic Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations for passive linear multicomponent media. Moreover, we discuss the connection of this map to Herglotz functions for isotropic and anisotropic multicomponent composites., Comment: 33 pages, 2 figures; this is a copy of chapter 4 in the book "Extending the Theory of Composites to Other Areas of Science" edited by Graeme W. Milton, 2016, Milton-Patton Publishers, ISBN: 978-1-4835-6919-2

We say that a complex number $\lambda$ is an extended eigenvalueof a bounded linear operator T on a Hilbert space H if there exists anonzero bounded linear operator X acting on H, called extended eigen-vector associated to $\lambda$, and satisfying the equation T X = $\lambda$XT . In thispaper we describe the sets of extended eigenvalues and extended eigen-vectors for the product of a positive and a self-adjoint operator whichare both injective. We also treat the case of normal operators.

Le but de cet article est d'obtenir la cyclicite de certaines classes de fonctions pour des operateurs de Toeplitz anti-analytique associes a un produit fini de Blaschke dans les espaces $H^p$ ou $1

An equivalent formulation of the von Neumann inequality states that the backward shift $S^*$ on $\ell_{2}$ is extremal, in the sense that if $T$ is a Hilbert space contraction, then $\|p(T)\| \leq \|p(S^*)\|$ for each polynomial $p$. We discuss several results of the following type : if $T$ is a Hilbert space contraction satisfying some constraints, then $S^*$ restricted to a suitable invariant subspace is an extremal operator. Several operator radii are used instead of the operator norm. Applications to inequalities of coefficients of rational functions positive on the torus are given., Comment: Preprint version

Let $M$ be a finite von Neumann algebra. In the first part, we give asymptotic results about $M$-stable sequences of weak*-continuous mappings which are related with operators belonging to $M$. In the second part, we extend, by a shorter way, similarity results given in [CaFa2] to unbounded semigroups of operators contained in a finite von Neumann algebra., Comment: 18 pages