In the recent years the transmission eigenvalue problem has been extensively studied for non-absorbing media. In this paper we initiate the study of this problem for absorbing media. In particular we show that, in the case of absorbing media, transmission eigenvalues form a discrete set, exist for sufficiently small absorption and for spherically stratified media exist without this assumption. ...

The heat trace asymptotics are discussed for operators of Laplace type with Dirichlet, Robin, spectral, D/N, and transmittal boundary conditions. The heat content asymptotics are discussed for operators with time dependent coefficients and Dirichlet or Robin boundary conditions.

We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell’s equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero.

In this paper, we employ transformation operators and Levinson’s density formula to study the distribution of interior transmission eigenvalues for a spherically stratified media. In particular, we show that under smoothness condition on the index of refraction that there exist an infinite number of complex eigenvalues and there exist situations when there are no real eigenvalues. We also consi...

This paper provides a survey of recent developments on the transmission eigenvalue problem.

Based on differences in the permittivity of coal and limestone rock, a resonant cavity measurement technique is presented for the rapid determination of the proportions of coal and limestone in powdered mixtures. The ratio of the resonant frequency shift and the change in the transmission factor when mixed coal and limestone samples are inserted into the cavity provides a means for estimating t...

We study the so-called interior transmission problem using the T-coercivity approach. In particular, we prove that this problem, which appears when one is interested in the reconstruction of the support of an inclusion embedded in a homogeneous medium, is of Fredholm type and that so-called transmission eigenvalues form at most a discrete set. The simple technique we propose allows to treat cas...

We consider the interior transmission eigenvalue (ITE) problem that arises when scattering by inhomogeneous media is studied. The ITE problem is not self-adjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (as they approach z = 1). We obtain a Weyl type formula ...

Let Ω ⊆ R be a non-empty open set for which the Sobolev embedding H 0 (Ω) −→ L(Ω) is compact, and let V ∈ L∞(Ω) be a potential taking only positive real values and satisfying the asymptotics V (·) 〈·〉 for some α ∈ ]3,∞[. We establish the discreteness of the set of real transmission eigenvalues for both Schrödinger and Helmholtz scattering with these potentials.