We study time harmonic scattering for the Helmholtz equation in R n . We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the relative scattering (a.k.a. far field) operator has a trivial kernel and cokernel at every real wavenumber. IntroductionThe diffraction of light around corners and edges, and through slits, provided the first evidence for the wave nature of light. The diffraction patterns caused by plane waves incident on corners or edges and were among the first scattered waves to be calculated [14]. Geometric optics expansions for scattered waves [9] reveal the presence of scattered waves in regions where the simple theory of optics does not. Much of our understanding of classical electromagnetism is based on these patterns. This is why a stealth airplane is built to minimize the scattering from corners and edges.Although the single frequency inverse scattering problem has a unique solution, the wave scattered from a single incident wave does not contain enough information to determine an obstacle or a penetrable scatterer. In many cases, the same scattered wave might have been scattered by a scatterer supported on a smaller set. In this paper we will show that, a penetrable scatterer whose support contains a right angle corner as an extreme point of its convex hull

We consider the inverse source problem of a fixed wavenumber: study properties of an acoustic source based on a single far-or nearfield measurement. We show that nonradiating sources having a convex or non-convex corner or edge on their boundary must vanish there. The same holds true for smooth enough transmission eigenfunctions. The proof is based on an energy identity from the enclosure method and the construction of a new type of planar complex geometrical optics solution whose logarithm is a branch of the square root. The latter allows us to deal with non-convex corners and edges.

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $V\in L^\infty(\Omega)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v, w\in L^2(\Omega)$ and $k\in\mathbb{R}_+$, \[(\Delta+k^2)v= 0 \quad \text{in } \Omega,\] \[(\Delta+k^2(1+V))w= 0 \quad \text{in } \Omega,\] \[w-v \in H^2_0(\Omega), \quad \lVert v \rVert_{L^2(\Omega)}=1. \] We show that the transmission eigenfunctions $v$ and $w$ carry the geometric information of $\mathrm{supp}(V)$. Indeed, it is proved that $v$ and $w$ vanish near a corner point on $\partial \Omega$ in a generic situation where the corner possesses an interior angle less than $\pi$ and the potential function $V$ does not vanish at the corner point. This is the first quantitative result concerning the intrinsic property of transmission eigenfunctions and enriches the classical spectral theory for Dirichlet/Neumann Laplacian. We also discuss its implications to inverse scattering theory and invisibility.Comment: 17 pages, addendum at arxiv:1710.0808