Interior Transmission Eigenvalue Problem Research Articles

On the distribution of Born transmission eigenvalues in the complex plane

We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.

The interior transmission eigenvalue problem for an anisotropic medium by a partially coated boundary

The interior transmission eigenvalue problem for an anisotropic medium by a partially coated boundary

Wave Patterns inside Transparent Scatterers

It may happen that under a certain wave interrogation, a medium scatterer produces no scattering. In such a case, the scattering field is trapped inside the scatterer and forms a certain interior resonant mode. We are concerned with the behavior of the wave propagation inside a transparent scatterer. It turns out that the study can be boiled down to analyzing the interior transmission eigenvalue problem. For isotropic mediums, it is shown in a series of recent works that the transmission eigenfunctions possess rich patterns. In this paper, we show that those spectral patterns also hold for anisotropic mediums.

The interior transmission eigenvalue problem for elastic waves in media with obstacles

In this paper, we investigate the interior transmission eigenvalue problem for elastic waves propagating outside a sound-soft or a sound-hard obstacle surrounded by an anisotropic layer. This study is motivated by the inverse problem of identifying an object embedded in an inhomogeneous media in the presence of elastic waves. Our analysis of this non-selfadjoint eigenvalue problem relies on the weak formulation of involved boundary value problems and some fundamental tools in functional analysis.

On the inverse scattering from anisotropic periodic layers and transmission eigenvalues

ABSTRACT This paper is concerned with the inverse scattering and the transmission eigenvalues for anisotropic periodic layers. For the inverse scattering problem, we study the Factorization method for shape reconstruction of the periodic layers from near field scattering data. This method provides a fast numerical algorithm as well as a unique determination for the shape reconstruction of the scatterer. We present a rigorous justification and numerical examples for the factorization method. The transmission eigenvalue problem in scattering have recently attracted a lot of attentions. Transmission eigenvalues can be determined from scattering data and they can provide information about the material parameters of the scatterers. In this paper, we formulate the interior transmission eigenvalue problem and prove the existence of infinitely many transmission eigenvalues for the scattering from anisotropic periodic layers.

Unique reconstruction of the potential for the interior transmission eigenvalue problem for spherically stratified media

This work deals with the interior transmission eigenvalue problem for a spherically stratified medium supported in {x: |x| = r = b}, which can be formulated as −y″ + q(x)y = λ2y with boundary conditions y(0) = 0 = y′(1) cos(aλ) − y(1) sin(aλ)/λ, where a = b/B with B being the size of the wave speed, measured by an integral. We provide a necessary and sufficient condition for the existence issue by giving a new method that allows unique reconstruction of a potential of this Sturm–Liouville problem from the spectrum of the problem and the set of the norming constants corresponding to the real eigenvalues when 0 < a < 1; and from the spectrum together with one additional piece of information when a = 1. The method is based on Mittag-Leffler expansions which can help us to decompose an entire function of exponential type into two functions of more smaller exponential types. This decomposition provides us a well-suited situation for utilizing Levin–Lyubarski interpolation formula to reconstruct the potential for our problem.

Efficient methods of computing interior transmission eigenvalues for the elastic waves

We study the interior transmission eigenvalue problem for the elastic wave scattering in this paper. We aim to show the distribution of positive eigenvalues by efficient numerical algorithms. Here the elastic waves are scattered by the perturbations of medium parameters, which include the elasticity tensor C and the density ρ. Let us denote (C0,ρ0) and (C1,ρ1) the background and the perturbed medium parameters, respectively. We consider two cases of perturbations, C0=C1,ρ1≠ρ0 (case 1) and C0≠C1,ρ1=ρ0 (case 2). After discretizing the associated PDEs by FEM, we are facing the computation of generalized eigenvalues problems (GEP) with matrices of large size. These GEPs contain huge number of nonphysical zeros (for case 1) or nonphysical infinities (for case 2). In order to locate several hundred positive eigenvalues effectively, we then convert GEPs to suitable quadratic eigenvalues problems (QEP). We then implement a quadratic Jacobi-Davidson method combining with partial locking or partial deflation techniques to compute 500 positive eigenvalues.

Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary

The interior transmission eigenvalue problem plays a basic role in the study of inverse scattering problems for an inhomogeneous medium. In this paper, we consider the electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with conductive boundary. Our main focus is to understand the associated eigenvalue problem, more specifically to prove the transmission eigenvalues form a discrete set and show that they exist by employing a variety of variational techniques under various assumptions on the index of refraction.

Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem

This work deals with the interior transmission eigenvalue problem: $y'' + {k^2}\eta \left( r \right)y = 0$ with boundary conditions ${y\left( 0 \right) = 0 = y'\left( 1 \right)\frac{{\sin k}}{k} - y\left( 1 \right)\cos k},$ where the function $\eta(r)$ is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption for the square of the index of refraction $\eta(r)$. Moreover, we provide a uniqueness theorem for the case $\int_0^1\sqrt{\eta(r)}dr>1$, by using all transmission eigenvalues (including their multiplicities) along with a partial information of $\eta(r)$ on the subinterval. The relationship between the proportion of the needed transmission eigenvalues and the length of the subinterval on the given $\eta(r)$ is also obtained.

The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations

In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert–Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.

The inverse interior transmission eigenvalue problem with mixed spectral data

The inverse spectral problem for the interior transmission eigenvalue problem with the unit time is studied by given spectral data. The authors show that the refractive index on the whole interval can be uniquely determined by parts of its transmission eigenvalues together with the partial information on the refractive index. In particular, we pose and solve a new type of inverse spectral problems involving the interior transmission eigenvalue problem with complex transmission eigenvalues except for at most finite real eigenvalues.

A modified transmission eigenvalue problem for scattering by a partially coated crack

We consider the detection of changes in the surface impedance of a partially coated crack, which we infer from the shift in a target signature arising from a modified interior transmission eigenvalue problem. We study this problem in a general setting in which the properties of the scattering medium are encoded in a bounded linear operator T, and we provide sufficient conditions for T that imply desirable properties of the eigenvalues of this problem. We conclude by placing scattering by a partially coated crack into this general framework and investigating the sensitivity of the associated eigenvalues to changes in the surface impedance with a series of numerical examples.

The inverse scattering problem for a conductive boundary condition and transmission eigenvalues

ABSTRACTIn this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. In particular, we are interested in two problems that arise from this inverse problem: the inverse conductivity problem and the corresponding interior transmission eigenvalue problem. The inverse conductivity problem is to recover the conductive boundary parameter from the measured scattering data. We prove that the measured scatted data uniquely determine the conductivity parameter as well as describe a direct algorithm to recover the conductivity. The interior transmission eigenvalue problem is an eigenvalue problem associated with the inverse scattering of such materials. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and discreteness for the case of an absorbing media. Lastly, several numerical and analytical results support the theory and we show that the inside–outside duality method can be used to reconstruct the interior conductive eigenvalues.

On $T$-coercive interior transmission eigenvalue problems on compact manifolds with smooth boundary

We consider an interior transmission eigenvalue problem on two compact Riemannian manifolds with common smooth boundary. We assume that this problem is locally anisotropic type. Then we prove that the set of interior transmission eigenvalues forms a discrete subset of complex plane. Moreover, we also mention the interior transmission eigenvalue free region. In order to prove our results, we employ the so-called $T$-coercivity method.

Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking

Abstract This article is concerned with the invisibility cloaking in electromagnetic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. Our study is based on an interior transmission eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Maxwell–Herglotz approximation of the associated interior transmission eigenfunctions. We provide the mathematical design of the cloaking device and sharply quantify the cloaking performance.

On isotropic cloaking and interior transmission eigenvalue problems

This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in Cakoni et al. (Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Problems and Imaging, 6 (2012), 373–398). Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Herglotz approximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.

On Electromagnetic Scattering from a Penetrable Corner

This article is concerned with the time-harmonic electromagnetic (EM) scattering from a generic inhomogeneous medium. It is shown that if there is a right corner on the support of the medium, then it scatters every pair of incident EM fields, excluding a possible class of EM fields which are of very particular forms. That is, for every pair of admissible incident EM fields, the corresponding scattered wave fields associated to the medium scatterer cannot be identically vanishing outside the support of the medium. Indeed, we achieve the corner scattering result by establishing a stronger result, that shows the failure of the analytic extension across the corner of certain EM fields satisfying the so-called interior transmission eigenvalue problem. This extends the relevant study in [3] for the acoustic scattering governed by the Helmholtz equation to the electromagnetic case governed by the Maxwell system. Substantial new challenges arise from the corresponding extension from the scalar PDE to the system of PDEs. Our mathematical arguments combine the analysis for interior transmission eigenvalue problems associated to the Maxwell system; the derivation of novel orthogonality relation for the solutions of Maxwell systems; the construction of complex-geometrical-optics (CGO) solutions for the Maxwell system with new Lp-estimates (p > 6) on the remainder terms; and the proof of the non-vanishing property for the Laplace transform of vectorial homogeneous harmonic polynomials.

The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary

In this paper, we investigate the interior transmission eigenvalue problem for an inhomogeneous media with conductive boundary conditions. We prove the discreteness and existence of the transmission eigenvalues. We also investigate the inverse spectral problem of gaining information about the material properties from the transmission eigenvalues. In particular, we prove that the first transmission eigenvalue is a monotonic function of the refractive index n and boundary conductivity parameter , and obtain a uniqueness result for constant coefficients. We provide some numerical examples to demonstrate the theoretical results in three dimensions.

Distribution of transmission eigenvalues and inverse spectral analysis with partial information on the refractive index

In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as y′′(r) + k2η(r)y(r) = 0 endowed with boundary conditions , where the refractive index η(r) is positive and real. We obtain the distribution of transmission eigenvalues under assumptions that and one of these conditions that (i) η(1) ≠ 1, (ii) η(1) = 1,η′(1) ≠ 0, and (iii) η(1) = 1,η′(1) = 0,η″(1) ≠ 0, respectively. Moreover, in the case a = 1, we prove that if partial information on η(r) is known on subdomain, then only a part of eigenvalues can uniquely determine η(r) on the whole interval, and the relationship between the proportion of the missing eigenvalues and the subinterval of the known information on η(r) is revealed. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Interior transmission eigenvalues of a rectangle

The problem of scattering of acoustic waves by an inhomogeneous medium is intimately connected with so called inside–outside duality, in which the interior transmission eigenvalue problem plays a fundamental role. Here a study of the interior transmission eigenvalues for rectangular domains of constant refractive index is made. By making a nonstandard use of the classical separation of variables technique both real and complex eigenvalues are determined.