TY  - JOUR
A1  - Pieronek, Lukas
A1  - Kleefeld, Andreas
T1  - On trajectories of complex-valued interior transmission eigenvalues
JF  - Inverse problems and imaging : IPI
N2  - This paper investigates the interior transmission problem for homogeneous media via eigenvalue trajectories parameterized by the magnitude of the refractive index. In the case that the scatterer is the unit disk, we prove that there is a one-to-one correspondence between complex-valued interior transmission eigenvalue trajectories and Dirichlet eigenvalues of the Laplacian which turn out to be exactly the trajectorial limit points as the refractive index tends to infinity. For general simply-connected scatterers in two or three dimensions, a corresponding relation is still open, but further theoretical results and numerical studies indicate a similar connection.
KW  - Interior transmission problem
KW  - Eigenvalue trajectories
KW  - Complex-valued eigenvalues
Y1  - 2024
U6  - http://dx.doi.org/10.3934/ipi.2023041
SN  - 1930-8337 (Print)
SN  - 1930-8345 (Online)
VL  - 18
IS  - 2
SP  - 480
EP  - 516
PB  - AIMS
CY  - Springfield, Mo
ER  - 
TY  - JOUR
A1  - Ayala, Rafael Ceja
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
A1  - Pallikarakis, Nikolaos
T1  - Analysis of the transmission eigenvalue problem with two conductivity parameters
JF  - Applicable Analysis
N2  - In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work.
KW  - Transmission Eigenvalues
KW  - Conductive Boundary Condition
KW  - Inverse Scattering
Y1  - 2023
U6  - http://dx.doi.org/10.1080/00036811.2023.2181167
SN  - 0003-6811
PB  - Taylor & Francis
ER  -