Bifurcations in Interior Transmission Eigenvalues: Theory and Computation
For researchers in inverse scattering and spectral analysis, this work provides a theoretical and computational understanding of bifurcations in interior transmission eigenvalues, though it is an incremental advance.
The paper develops a theoretical framework for identifying non-smooth spectral behavior in interior transmission eigenvalue problems and uses a match-based adaptive contour eigensolver to track eigenvalue trajectories, confirming predictions with numerical experiments.
The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the corresponding spectral map from material parameters to eigenpairs may exhibit non-smooth or bifurcating behavior. In this work, we develop a theoretical framework identifying sufficient conditions for such non-smooth spectral behavior in the ITP on general domains. We further specialize our analysis to some radially symmetric geometries, enabling a more precise characterization of bifurcations in the spectrum. Computationally, we formulate the ITP as a parametric, discrete, nonlinear eigenproblem and use a match-based adaptive contour eigensolver to accurately and efficiently track eigenvalue trajectories under parameter variation. Numerical experiments confirm the theoretical predictions and reveal novel non-smooth spectral effects.