Finiteness of Nonscattering Wavenumbers for Herglotz Incident Waves

TL;DR

This paper proves that for certain inhomogeneous media and star-shaped domains, there are only finitely many wavenumbers that do not scatter incident waves, highlighting the role of symmetry and geometry in scattering phenomena.

Contribution

It establishes finiteness of nonscattering wavenumbers for star-shaped domains with inhomogeneous media, extending previous results and removing geometric restrictions.

Findings

Finiteness of nonscattering wavenumbers for elliptical domains with constant media.

Finiteness results for star-shaped domains with broader geometric assumptions.

Infinite nonscattering sequences are linked to exact radial symmetry and are unstable under perturbations.

Abstract

This paper continues the study initiated in [30] on nonscattering phenomena for inhomogeneous media. We investigate star-shaped domains in $\mathbb{R}^2$ and establish finiteness results for nonscattering wavenumbers associated with Herglotz incident waves of fixed density. First, for ellipses with constant medium coefficient $q\in(0,1)\cup(1,\infty)$, we prove that there exist at most finitely many nonscattering wavenumbers. This generalizes and strengthens the corresponding results in [30], in particular removing additional geometric restrictions in the case $q>1$. Second, for admissible $C^2$ star-shaped domains with $q\in(0,1)$, we establish analogous finiteness results under broader geometric assumptions on the radius function. Our results reveal that infinite sequences of nonscattering wavenumbers are tied to exact radial symmetry and cannot persist under admissible geometric perturbations.