Corners Always Scatter

Abstract

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

Strictly convex corners scatter

“…It will be important to have L q estimates for large q with suitable decay for the remainder term ψ in these solutions. In [BPS14], such solutions were constructed in all dimensions n ≥ 2 but for "polygonal" cones C (that is, the cross-section of the cone is a polygon). For our higher dimensional result it is more convenient to use circular cones, and it turns out that the argument of [BPS14] is not valid in this case.…”

“…The recent work [BPS14] suggests that corner points in the scattering potential always generate a scattered wave. More precisely, [BPS14] proves the absence of non-scattering energies in acoustic scattering for 1 certain contrasts having corners with a right angle, i.e.…”

“…Structure of argument. We follow the approach of [BPS14] and argue by contradiction. Assume that λ ∈ R + is a non-scattering energy for the potential V .…”

“…Up to this point, we have closely followed the approach of [BPS14]. We now depart from this approach and move to polar coordinates in the Laplace transform identity.…”

“…The main difference between our approach and [BPS14] is that we will perform computations in terms of these spherical harmonics. In Section 5 we discuss the case n = 2, which is particularly simple.…”