Bifurcations in Interior Transmission Eigenvalues: Theory and Computation

TL;DR

This paper investigates bifurcations in the spectrum of the interior transmission eigenvalue problem, providing theoretical conditions, specialized analysis for symmetric geometries, and computational methods to track eigenvalue behavior.

Contribution

It offers a new theoretical framework for understanding spectral bifurcations in ITP and introduces an efficient computational approach for eigenvalue tracking.

Findings

Theoretical conditions for spectral bifurcations in ITP.

Specialized analysis for radially symmetric geometries.

Numerical experiments confirm non-smooth spectral effects.

Abstract

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.