Transmission Eigenvalues

Abstract

Abstract. The scattering of a time-harmonic plane wave in an inhomogeneous medium is modeled by the scattering problem for the Helmholtz equation. A transmission eigenvalue is a wavenumber at which the scattering operator has a non-trivial kernel or cokernel. Because many sampling methods for locating scatterers succeed only at wavenumbers that are not transmission eigenvalues, they have been studied for some time. Nevertheless, the existence of transmission eigenvalues has previously been proved only for radial scatterers. In this paper, we prove existence for scatterers without radial symmetry.

A Spectral-Element Method for Transmission Eigenvalue Problems

“…Recently, Kirsch [14] proved some existence results for the transmission eigenvalue problem (1.1)-(1.4) (see also [6] for some related results). While the result in [14] can not be directly applied to the one dimensional problems (2.6)-(2.8) and (2.9)-(2.11), we can adapt the method in [14,18] to prove an existence result.…”

“…Hence, k is a transmission eigenvalue of (2.6)-(2.8) with (2.7) in the sense of (2.17). In order to use the argument in [14,18] to prove the existence of at least one transmission eigenvalue, an essential step is to constructv ∈ H 2…”

“…The existence and computation of transmission eigenvalues have been studied recently by many researchers [6][7][8]13,14,18,19]. In particular, several finite-element methods are presented in [8].…”

Surface integral formulation of the interior transmission problem

“…The theory of inverse scattering for acoustic and electromagnetic waves, is an active area of research with significant developments in the past few years and more specifically the so-called interior transmission problem (ITP) and its transmission eigenvalues [3,11,12,19,22,8,4,6,9,15,17,7,21]. Although simply stated, the interior transmission problem is not covered by the standard theory of elliptic partial differential equations since as it stands it is neither elliptic nor self-adjoint.…”

Strictly convex corners scatter

“…This has led people to study the usually larger class of interior transmission eigenvalues which first appeared in [CM88,Ki86]. For acoustic scattering, the transmission eigenvalues often form an infinite discrete set [PS08,CGH10], and in recent years they have been studied intensively. For more information about transmission eigenvalues, we recommend the survey [CH13a] as well as the articles mentioned in the recent editorial [CH13b] and their references.…”