Higher-order necessary conditions for nonscattering

We study the scattering of incident waves by a class of planar analytic inhomogeneities with constant refractive index, governed by the Helmholtz equation. We derive an infinite hierarchy of necessary conditions for nonscattering: any complex analytic nonscattering incident wave must satisfy infinitely many differential constraints at a specific point determined by the geometry of the inhomogeneity. As an application, we show that for any noncircular elliptical inhomogeneity, every single separated (Mathieu) mode necessarily scatters. This is in stark contrast to the circular case, where single separated (Bessel) modes are nonscattering at infinitely many wavenumbers.