The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature.
This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $π$. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures e…
We present a comprehensive study of new discoveries on the spectral patterns of elastic transmission eigenfunctions, including boundary localisation, surface resonance, and stress concentration. In the case where the domain is radial and the underlying parameters are constant, we give rigorous justifications and derive a thorough understanding of those intriguing geometric and physical patterns. We also present numerical examples to verify that the same results hold in general geometric and parameter setups.
In this addendum, we relax a restrictive assumption in [1] needed for the interior transmission eigenfunctions to hold the intrinsic geometric vanishing property in a corner. In addition we present in more detail another assumption which can also guarantee the vanishing property, namely being locally $H^2$ near the corner. This was mentioned briefly in [1].
Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.
The transmission eigenvalue problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for a system without complementing conditions, i.e., the two equations of the system have the same coefficients for the second order terms, and thus being degenerate. These coefficients are allowed to be anisotropic and are assumed to be of class $C^2$. One of the keys of the analysis is to establish the well-posedness and the regula…
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the c…
We consider the inverse source problem of a fixed wavenumber: study properties of an acoustic source based on a single far- or near-field measurement. We show that nonradiating sources having a convex or non-convex corner or edge on their boundary must vanish there. The same holds true for smooth enough transmission eigenfunctions. The proof is based on an energy identity from the enclosure method and the construction of a new type of planar complex geometrical optics solution whose logarithm is a branch of the square root. The latter allows us to deal with non-convex corners and edges.
Consider the nonlinear and completely continuous scattering map \[ \mathcal{S}\big((Ω; λ, μ, V), \mathbf{u}^i\big)=\mathbf{u}_t^\infty(\hat{\mathbf{x}}), \quad \hat{\mathbf{x}}\in\mathbb{S}^{n-1}, \] which sends an inhomogeneous elastic scatterer $(Ω; λ, μ, V)$ to its far-field pattern $\mathbf{u}_t^\infty$ due to an incident wave field $\mathbf{u}^i$ via the Lamé system. Here, $(λ, μ, V)$ signifies the medium configuration of an elastic scatterer that is compactly supported in $Ω$. In this paper, we are concerned with the intrinsic geometric structure of the kernel space of $\mathcal{S}$, whi…
We consider several intriguingly connected topics in the theory of wave propagation: geometrical characterizations of radiationless sources, non-radiating incident waves, interior transmission eigenfunctions, and their applications to inverse scattering. Our major novel discovery is a localization and geometrization property. We first show that a scatterer, which might be an active source or an inhomogeneous index of refraction, cannot be completely invisible if its support is small compared to the wavelength and scattering intensity. Next, we localize and geometrize the "smallness" results to…
In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to…
Let $Ω$ be a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $V\in L^\infty(Ω)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v, w\in L^2(Ω)$ and $k\in\mathbb{R}_+$, \[(Δ+k^2)v= 0 \quad \text{in } Ω,\] \[(Δ+k^2(1+V))w= 0 \quad \text{in } Ω,\] \[w-v \in H^2_0(Ω), \quad \lVert v \rVert_{L^2(Ω)}=1. \] We show that the transmission eigenfunctions $v$ and $w$ carry the geometric information of $\mathrm{supp}(V)$. Indeed, it is proved that $v$ and $w$ vanish near a corner point on $\partial Ω$ in a generic situation where the corner possesses an int…
Consider the transmission eigenvalue problem for $u \in H^1(Ω)$ and $v\in H^1(Ω)$ associated with $(Ω; σ, \mathbf{n}^2)$, where $Ω$ is a ball in $\mathbb{R}^N$, $N=2,3$. If $σ$ and $\mathbf{n}$ are both radially symmetric, namely they are functions of the radial parameter $r$ only, we show that there exists a sequence of transmission eigenfunctions $\{u_m, v_m\}_{m\in\mathbb{N}}$ associated with $k_m\rightarrow+\infty$ as $m\rightarrow+\infty$ such that the $L^2$-energies of $v_m$'s are concentrated around $\partialΩ$. If $σ$ and $\mathbf{n}$ are both constant, we show the existence of transmi…
In this paper, we study an inverse problem of determining the cross section of an infinitely long cylindrical-like material structure from the transverse electromagnetic scattering measurement. We establish a sharp logarithmic stability result in determining a polygonal scatterer by a single far-field measurement. The argument in establishing the stability result is localized around a corner and can be as well used to produce two highly intriguing implications for invisibility and transmission resonance in the wave scattering theory. In fact, we show that if a generic medium scatterer possesse…
Transmission eigenfunctions are certain interior resonant modes that are of central importance to the wave scattering theory. In this paper, we present the discovery of novel global rigidity properties of the transmission eigenfunctions associated with the Maxwell system. It is shown that the transmission eigenfunctions carry the geometrical and topological information of the underlying domain. We present both analytical and numerical results of these intriguing rigidity properties. As an interesting application, we propose an illusion scheme of artificially generating a mirage image of any gi…
We construct a specific family of eigenfunctions for a Laplace operator with coefficients having a jump across an interface. These eigenfunctions have an exponential concentration arbitrarily close to the interface, and therefore could be considered as whispering gallery modes. The proof is based on an appropriate Agmon estimate. We deduce as a corollary that the quantitative unique continuation result for waves propagating in singular media proved by the author in [Fil22] is optimal.
We present the discovery of a novel and intriguing global geometric structure of the (interior) transmission eigenfunctions associated with the Helmholtz system. It is shown in generic scenarios that there always exists a sequence of transmission eigenfunctions with the corresponding eigenvalues going to infinity such that those eigenfunctions are localized around the boundary of the domain. We provide a comprehensive and rigorous justification in the case within the radial geometry, whereas for the non-radial case, we conduct extensive numerical experiments to quantitatively verify the locali…
Reconstruction via the intrinsic geometric structures of interior transmission eigenfunctions
Southern University of Science and Technology（南方科技大学）；Inha University（仁荷大学）；Hong Kong Baptist University（香港浸会大学）；HKBU Institute of Research and Continuing Education（香港浸会大学研究与继续教育学院）
We consider the transmission eigenvalue problem for an impenetrable obstacle with Dirichlet boundary condition surrounded by a thin layer of non-absorbing inhomogeneous material. We derive a rigorous asymptotic expansion for the first transmission eigenvalue with respect to the thickness of the thin layer. Our convergence analysis is based on a Max-Min principle and an iterative approach which involves estimates on the corresponding eigenfunctions. We provide explicit expressions for the terms in the asymptotic expansion up to order three.
To $1/2$-logconcavity and beyond: Geometric properties of Dirichlet eigenfunctions
Institute of Mathematics, Hunan University（湖南大学数学研究所）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；School of Mathematical Sciences, Soochow University（苏州大学数学科学学院）
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. 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 noti…
In this paper, we present a Spectral-Galerkin Method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl's law.…
In this paper we prove some results on interior transmission eigenvalues. First, under rea- sonable assumptions, we prove that the spectrum is a discrete countable set and the generalized eigenfunctions spanned a dense space in the range of resolvent. This is a consequence of spectral theory of Hilbert-Schmidt operators. The main ingredient is to prove a smoothing property of resolvent. This allows to prove that a power of the resolvent is Hilbert-Schmidt. We obtain an estimate of the number of eigenvalues, counting with multiplicities, with modulus less than t2 when t is large. We prove also …
We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical s…
This paper is concerned with the inverse electromagnetic scattering problem for anisotropic media. We use the interior resonant modes to develop an inverse scattering scheme for imaging the scatterer. The whole procedure consists of three phases. First, we determine the interior Maxwell transmission eigenvalues of the scatterer from a family of far-field data by the mechanism of the linear sampling method. Next, we determine the corresponding transmission eigenfunctions by solving a constrained optimization problem. Finally, based on both global and local geometric properties of the transmissi…
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 \cite{CCH}. 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 m…
In this paper, we utilize Fokas method to investigate the initial-boundary value problems (IBVPs) of the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation on the half-line, which can simulate the nonlinear transmission and interaction of ultrashort pulses in the high-speed optical fiber transmission system, and describe the nonlinear spin excitation phenomenon of one-dimensional Heisenberg ferromagnetic chain with eight poles and dipole interaction. By discussing the eigenfunctions of Lax pair of FODNLS equation and the analysis and symmetry of the scattering matrix, the IBVPs of…
