Finiteness of Nonscattering Wavenumbers for Herglotz Incident Waves

This paper continues the study initiated in \cite{VogXia25} 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 we establish finiteness for all constant contrasts $q\neq 1$, removing the geometric restrictions required in previous work. Second, for admissible star-shaped domains with $q\in(0,1)$, we introduce a flexible interval-wise geometric framework that unifies and generalizes earlier finiteness results. Our results reveal that infinite sequences of nonscattering wavenumbers are tied to exact radial symmetry and cannot persist under admissible geometric perturbations. %extend those of \cite{VogXia25} to the regime $0<q<1$ and reveal a rigidity phenomenon for nonscattering behavior beyond the radially symmetric setting.