Finiteness of Nonscattering Wavenumbers for Herglotz Incident Waves

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.