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

Abstract

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 η, and obtain a uniqueness result for constant coefficients. We provide some numerical examples to demonstrate the theoretical results in three dimensions.

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

“…So we begin by considering the case of an absorbing material with complex valued conductivity. It is shown numerically that the transmission eigenvalues can be determined from the FF-data in [1] for a non-absorbing material and following the analysis in [5] one can prove the following results using either the far field or near field operator, defined in (11) and (10) respectively. Theorem 3.1.…”

“…Here we let h (1) p denote the spherical Hankel function of the first kind of order p and Y m p is the spherical harmonic. Applying the boundary condition on ∂D gives the 2 × 2 system…”

“…This eigenvalue problem can be derived from considering an obstacle that 'does not' scatter for a specific wave number k and incident field. The interior eigenvalue problem for an inhomogeneous media with a conductive boundary was studied in [1], where they used the theoretical results in [8] to prove existence of real eigenvalues. Their numerical experiments seem to indicate that the interior conductive eigenvalues can be reconstructed using the linear sampling method and that they converge to the interior transmission eigenvalues as the conductivity parameter goes to zero.…”

Analysis of two transmission eigenvalue problems with a coated boundary condition

“…See the manuscripts [9] for a detailed account of the main results and techniques for these eigenvalue problems for the scalar scattering problem. First, we will study the problem for the electromagnetic scattering that is analogous to the problem studied in [6] and [20]. Next, being motivated by the new eigenvalue problems studied in [4] and [5] we will consider the 'zero-index' transmission eigenvalue problem for the scalar scattering problem.…”

Approximation of the Zero-Index Transmission Eigenvalues with a Conductive Boundary and Parameter Estimation

“…Next, is the fact that they are linear eigenvalue problems. This zero-index eigenvalue problem with a conductive boundary condition was introduced in [21] and was motivated by the work in [8,22] for the classical transmission eigenvalue problem with a conductive boundary and [6] for the scattering problem without a conductive boundary. We are also interested in the inverse spectral problem of estimating the refractive index with little a prior knowledge of the boundary conductivity parameter.…”