Bifurcations in Interior Transmission Eigenvalues: Theory and Computation

TL;DR

This work investigates how interior transmission eigenvalues (ITEs) depend on the refractive index in inverse scattering. It develops a general theory identifying sufficient conditions for non-smooth spectral behavior and applies it to radially symmetric domains (disk and annulus), where separation of variables yields explicit nonlinear eigenproblems. The authors recast the ITP as a parametric nonlinear eigenproblem and employ a contour-based Beyn method with the match-based adaptive contour eigensolver (MACE) to track eigenvalue trajectories under parameter variation, revealing novel non-smooth phenomena. Theoretical results are complemented by detailed disk and annulus analyses and extensive numerical experiments that validate bifurcation scenarios (e.g., cubic bifurcations at Bessel zeros for the disk and mode-veering in the annulus). The work provides a robust computational framework for studying parameter-dependent ITPs and lays groundwork for multi-parameter extensions and non-radial geometries.

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.

Figures (11)

• Figure 1: Examples of non-smooth eigenpair behavior. We set $\varepsilon=0.05$ in the two right-hand-side plots.
• Figure 1: Seven ITE trajectories (five of which are purely real) for the unit disk in the parameter range $p\in[1.01,32]$. Arrows are used to indicate the direction of travel along the dotted magenta curve with triangular markers. Real eigenvalues travel from right to left. The symbol "$\otimes$" marks the position of the Laplacian eigenvalue $\kappa^\star$.
• Figure 1: Ten ITE trajectories (eight of which are purely real) with $m=0$ for the annulus with $r=0.1$ in the parameter range $p\in[6,64]$. The arrow is used to indicate the direction of travel along the dotted magenta curve with triangular markers. The symbol "$\otimes$" marks the position of the Laplacian eigenvalue $\kappa^\star$.
• Figure 2: Real (left) and imaginary (center) parts of the ITE trajectories for the unit disk. The indicator $I$ is included in the right plot.
• Figure 2: Real (left) and imaginary (center) parts of the ITE trajectories for the annulus with $m=0$ and $r=0.1$. The indicator $\overline{I}$ is included in the right plot.

Theorems & Definitions (27)

• Remark 2.1
• Definition 3.1: See, e.g., Chapter 2 of kato
• Remark 3.2
• Theorem 3.3: See, e.g., Theorem 2.1 in AnChLa93 and Section 2.1.2 in kato
• Definition 3.4
• Theorem 4.1: Corollary 2.3, Lemma 2.4, Lemma 2.6, and Theorem 2.12 in PieKle24
• Theorem 4.2
• Proof 1
• Corollary 4.3
• Proof 2