We consider an acoustic waveguide modeled as follows:

$ \left \{\begin {matrix}
\Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\
\frac{\partial u}{\partial y}=0& on & \partial \Omega
\end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped modes. Associated eigenvalues $\lambda = k^2$ are embedded in the essential spectrum $\mathbb{R}^+$. They can be computed as the real part of the complex spectrum of a non-self-adjoint eigenvalue problem, defined by using the so-called Perfectly Matched Layers (which consist in a complex dilation in the infinite direction) [1].
We show here that it is possible, by modifying in particular the parameters of the Perfectly Matched Layers, to define new complex spectra which include, in addition to trapped modes, frequencies where the potential $V$ is, in some sense, invisible to one incident wave.
Our approach allows to extend to higher dimension the results obtained in [2] on a 1D model problem.[-]

Modal analysis using Quasi-Normal Modes (QNM) is now an essential tool to interpret the behavior of open photonic devices based on their intrinsic resonances. When dealing with frequency dispersive media, which is unavoidable in optics, the eigenvalue problem to be considered becomes non-linear due to the dependence of the permittivity with respect to the eigenvalue. The same situation occurs when dealing with frequency dependent boundary conditions at infinity, such as dispersive Perfectly Matched Layers (PMLs), Absorbing Boundary Conditions (ABC), or higher order approximations of the Dirichletto-Neumann operator. In this talk, we will present the various numerical tools recently introduced allowing to perform the QNM expansion with dispersive media in open geometries: (i) Starting with a general causal rational function as a permittivity model [1], (ii) we will review various linearization schemes to tackle the non-linear eigenvalue problem using finite elements, (iii) to finally show a general frame based on the Keldysh theorem to expand [3] the solution of direct problems.
In collaboration with Carmen Campos (Universitat Politècnica de València), Christophe Geuzaine (University of Liège) , Boris Gralak (Institut Fresnel), André Nicolet (Institut Fresnel), Jose E. Roman (Universitat Politècnica de València), Frédéric Zolla (Institut Fresnel).[-]

Various techniques based on Herglotz-Nevanlinna functions and optimization combined with duality have successfully been used to determine bounds for passive electromagnetic systems. Although these methods are very powerful and can be applied to large classes of problems, there are still challenges. In this presentation, we focus on the properties of Herglotz-Nevanlinna functions and optimization formulations and how they are applied to bandwidth bounds for antenna and scattering problems.[-]

Lamination of two materials with tensors $L_{1}$ and $L_{2}$ generates an effective tensor $L_{*}$. For certain fractional linear transformations $W(L)$, dependent on the property under consideration (conduction, elasticity, thermoelasticity, piezoelectricity, poroelasticity, etc.) and on the direction of lamination, $W\left(L_{*}\right)$ is just a weighted average of $W\left(L_{1}\right)$ and $W\left(L_{2}\right)$, weighted by the volume fractions occupied by the two materials, and this gives $L_{*}$ in terms of $L_{1}$ and $L_{2}$. Given an original set of materials one may laminate them together iteratively on larger and larger widely separated length scales, at each stage possibly laminating together two materials both of which are already laminates. Ultimately, one gets a hierarchical laminate with the lamination process represented by a tree with the leaves corresponding to the original set of materials, and with the volume fractions and direction of lamination specified at each vertex. It is amazing to see the range of effective tensors $L_{*}$ one can obtain, or effective tensor functions $L_{*}\left(L_{1}, L_{2}\right)$ one can obtain if, say, there are just two original materials. These functions $L_{*}\left(L_{1}, L_{2}\right)$ are closely related to Herglotz-Nevanlinna-Stieltjes functions. Here we will survey many results, some old, some surprising, on what effective tensors, and effective tensor functions, can be obtained lamination. In some cases the effective tensor or effective tensor function of any microgeometry can be mimicked by one of a hierarchical laminate. The question of what can be achieved is closely tied to the classic problem of rank-1 convexification and whether and when it equals quasiconvexification.[-]

A Herglotz-Nevanlinna function is a holomorphic function $f$, defined in the upper half-plane $\mathbb{H}:=\{z \in \mathbb{C} \mid \Im z>0\}$, such that $\Im f(z) \geq 0$ for all $z \in \mathbb{H}$, and they are the functions in focus at the present conference. These functions are also called Pick functions, and they are characterized as the functions of the form$$f(z)=\alpha z+\beta+\int_{-\infty}^{\infty} \frac{t z+1}{t-z} d \tau(t), \quad z \in \mathbb{H}$$where $\alpha \geq 0, \beta \in \mathbb{R}$ and $\tau$ is a positive finite measure on $\mathbb{R}$.
Since $\mathbb{H}$ is a simply connected domain, caracterization of this class of functions is the same as characterization of the set of non-negative harmonic functions in $\mathbb{H}$ and by conformal mapping this set is in one-to-one correspondence with the set of non-negative harmonic functions in the unit disc.
We shall discuss various subclasses of Pick functions and their relation to other important classes of functions such as the completely monotonic functions and the subclass of Stieltjes functions. We recall that these classes are the functions $f:] 0, \infty[\rightarrow \mathbb{R}$ of the form$$f(x)=\int_{0}^{\infty} e^{-s x} d \mu(s), \quad \text { resp. } f(x)=a+\int_{0}^{\infty} \frac{d \mu(s)}{x+s}$$where $a \geq 0$ and $\mu$ is a non-negative measure on $[0, \infty[$.At the 7 th OPSFA, Copenhagen 2003 , we posed the problem of determining the largest value $\alpha=\alpha^{*}>0$ for which $f_{\alpha}(x)=e^{\alpha}-(1+1 / x)^{\alpha x}, x>0$ is a completely monotonic function, and it was noticed that $1 \leq \alpha^{*}<3$ and that graphs suggested that $\alpha^{*}>2$. The value has now been calculated with 20 decimals starting with $\alpha^{*} \approx 2.29965$.This is based on a recent result obtained in collaboration with Massa and Peron from Brazil. We have found a family $\varphi_{\alpha}, \alpha>0$ of entire functions such that$$f_{\alpha}(x)=\int_{0}^{\infty} e^{-s x} \varphi_{\alpha}(s) d s, \quad x>0 .$$We showed that each function $\varphi_{\alpha}$ has an alternating power series expansion, whose coefficients are determined as an explicit sequence of polynomials in $\alpha$. It is therefore possible to calculate as accurately as desired for which values of $\alpha$ the function $\varphi_{\alpha}$ is non-negative on $\left[0, \infty\left[\right.\right.$. It turned out that the functions $\varphi_{\alpha}$ are 'close' to the well known Bessel function $J_{1}$ when $\alpha$ is large, and 'close' to the Lambert $W$ function, when $\alpha$ is small.[-]

In collaboration with Maxence Cassier (Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel) and Luis Alejandro Rosas Martinez (POEMS, CNRS, INRIA, ENSTA Paris).
It is well-known that electromagnetic dispersive structures such as metamaterials obey mathematical models whose construction, based on fundamental physical such as causality and passivity, emphasizes the role of Herglotz functions. Among these models an important class is provided by generalized Drude-Lorentz models, see e.g. [1]. In this work, we are interested in dissipative Drude-Lorentz open structures and 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 two different approaches, one based on (frequency dependent) Lyapounov estimates and the other on modal analysis, we show that this decay is polynomial in time. These results generalize a part the ones obtained for bounded media in [2] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approaches developed here is to be directly connected to the physics of the system via energy balances or modes behavior.[-]

In collaboration with Patrick Joly (Poems, Cnrs, Inria, Ensta Paris) and Adrien Semin (Technische Universität Darmstadt).
We consider the problem of a numerical resolution of a weighted wave equation on a fractal tree, which originates from an asymptotic modelling of sound propagation in lungs. Performing numerical simulations for this model relies on truncation of the computational domain to a finite tree. This is done by imposing a Dirichlet-to-Neumann (DtN) boundary condition at the ends of the truncated tree. The symbols of the DtN operators are Herglotz-Nevanlinna functions, and their Herglotz-Nevanlinna property is intimately related to the stability of the problem. Because in practical simulations we have to approximate the DtN operators, the stability of the approximated problem can be ensured if the approximated symbols of the DtNs are Herglotz-Nevanlinna functions as well. One such approximation is given by truncating to finitely many terms the meromorphic series representing the symbol of the DtN. We present the convergence analysis of this method, which relies on the estimates on the counting function for the weighted Laplacian and on the normal traces of the eigenfunctions. We finish the discussion with numerical experiments.[-]

In this survey talk we are first going to recall several equivalent (but partly still quite different) ways to characterize the class of Herglotz-Nevanlinna functions, and discuss the corresponding representations and tools. Based on this point of view there are different ways to extend the class of Herglotz-Nevanlinna functions by generalizing the defining properties. In this talk we are going to discuss few of them, with special focus on wether classical tools are still available or where they can fail.[-]

In collaboration with Graeme W. Milton (University of Utah) and Mihai Putinar (University of California at Santa Barbara).
In this talk, we will discuss bounds on the quasistatic response of lossy two-phase materials when subject to applied electromagnetic fields (or antiplane elastic fields), that can have any variation in time, not necessarily a fixed frequency one. Interestingly enough, appropriate choices of the applied field can directly provide the volume fractions of the phases from measurements at specific times and, for specially tailored boundary conditions, this occurs at any time. We also show how time varying boundary conditions, not oscillating at a single frequency, can be designed to exactly retrieve the response at a single frequency.[-]