On trajectories of complex-valued interior transmission eigenvalues

TL;DR

This paper studies the behavior of complex eigenvalue trajectories in the interior transmission problem, revealing their connection to Dirichlet eigenvalues and proposing a conjecture for general shapes.

Contribution

It provides a theoretical analysis of eigenvalue trajectories for the interior transmission problem and uncovers a numerical correspondence with Dirichlet eigenvalues for various scatterers.

Findings

Eigenvalue trajectories intersect the real axis only at Dirichlet eigenvalues.

Trajectorial limits as refractive index tends to infinity are Dirichlet eigenvalues.

Numerical experiments suggest a one-to-one correspondence between eigenvalue trajectories and Dirichlet eigenvalues.

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.