On trajectories of complex-valued interior transmission eigenvalues

TL;DR

The paper investigates complex-valued interior transmission eigenvalue (ITE) trajectories for the acoustic interior transmission problem in homogeneous media, revealing a deep link to Dirichlet Laplacian eigenvalues (DELs). It provides a rigorous analysis for the unit disk, showing that non-real ITE trajectories satisfy $F_p(n,\kappa)=0$ and that all accumulation points as $n\to\infty$ are DELs, with a one-to-one correspondence between complex-conjugate ITE trajectories and DELs, including multiplicities. Complementing the theory, extensive numerical experiments across convex and non-convex, 2D and 3D scatterers (disk, ball, ellipse, square, triangle, clover, deformed shapes, ellipsoid) demonstrate that complex ITE trajectories converge to DELs and that the accumulation behavior persists generally for simply-connected domains, supporting a broad conjecture. The results illuminate an intrinsic relation between non-real ITEs and DELs, with implications for inverse scattering and spectral theory, and point to a general conjecture: for any bounded simply-connected scatterer, there is a one-to-one DEL–ITE trajectory correspondence preserving multiplicities as $n\to\infty$.

Abstract

This paper investigates properties of complex-valued eigenvalue trajectories for the interior transmission problem parametrized by the index of refraction for homogeneous media. Our theoretical analysis for the unit disk shows that the only intersection points with the real axis, as well as the unique trajectorial limit points as the refractive index tends to infinity, are Dirichlet eigenvalues of the Laplacian. Complementing numerical experiments even give rise to an underlying one-to-one correspondence between Dirichlet eigenvalues of the Laplacian and complex-valued interior transmission eigenvalue trajectories. We also examine other scatterers than the disk for which similar numerical observations can be made. We summarize our results in a conjecture for general simply-connected scatterers.

Figures (13)

• Figure 1: Snapshot of the holomorphic flow (red arrows) generated by $\kappa\mapsto d_n(\kappa)/|d_n(\kappa)|$ for $n=4$ and $p=0$. The plot is to illustrate that corresponding solutions, including ITE trajectories, cannot approach or escape the real axis tangentially from or into the complex plane near DELs (blue asterisk), respectively.
• Figure 2: Illustration of the sectors $L_\gamma$, $M_\gamma$, $R_\gamma$ which do not contain any non-real ITEs if the refractive index is such that the complex-conjugated pair of complex-valued ITE trajectories intersect the DEL $\kappa^\ast$. For $M_\gamma$ (red) this fact is shown by exploiting Rouché's theorem in combination with \ref{['frac']}, while for $L_\gamma$ or $R_\gamma$ (green) one uses that any complex-valued solution to \ref{['dkappa']} (orange) would have hit the real axis or the dashed boundary line, both of which is excluded for ITE trajectories.
• Figure 3: The first three complex-conjugated pairs of complex-valued ITE trajectories for the unit disk and $p\in\{0,1,2\}$ without counting multiplicity, respectively, using $n\in (1,16]$.
• Figure 4: The first complex-conjugated pair of complex-valued ITE trajectories for the unit disk and $p=0$ from \ref{['fig1']} extended to $n\in (0,16]$ (color bar refers to $n<1$ only).
• Figure 5: The first three complex-conjugated pairs of complex-valued ITE trajectories for the unit ball and $p\in\{0,1,2\}$ without counting multiplicity, respectively, using $n\in (1,16]$.

Theorems & Definitions (25)

• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof