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 [Formula: see text] 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 [Formula: see text], removing the geometric restrictions required in previous work. Second, for admissible star-shaped domains with [Formula: see text], 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.