Strictly convex corners scatter

Abstract

Abstract. We prove the absence of non-scattering energies for potentials in the plane having a corner of angle smaller than π. This extends the earlier result of Blåsten, Päivärinta and Sylvester who considered rectangular corners. In three dimensions, we prove a similar result for any potential with a circular conic corner whose opening angle is outside a countable subset of (0, π).

Non-scattering energies and transmission eigenvalues in H^n

“…Then standard arguments in corner scattering imply that ϕ(x 0 ) = 0. In more detail, the arguments of Section 5 in [33] apply verbatim in the two dimensional case, and show that the Laplace transform cannot vanish for all admissible ρ 0 . Similarly, in the three and higher dimensional cases where C is a right-angled corner, Theorem 2.5 of [6] imply the same statement.…”

“…We will choose suitable coordinates in H n and conjugate the free operator H 0 with a suitable function to show the existence of these solutions. This allows us to bring past techniques of [6,33,20] into the hyperbolic setting. By conjugating the free operator H 0 from (1) on page 4 with a suitable function K we get…”

“…We are going to apply techniques from [6,33,20] to construct complex geometrical optics solutions u 0 . We start with the equation (H 0 +λ ν V −λ)u 0 = 0 in H n .…”

“…The remaining steps involve choosing a suitable set of coordinates for the hyperbolic space under which the corner of interest looks like a Euclidean corner, and then estimating the decay rates of various terms involved in the orthogonality relation. This reduces the problem to showing that the Laplace transform of the product of the characteristic function of a cone and a nontrivial harmonic polynomial cannot vanish identically on the complex characteristic manifold of points ρ ∈ C n satisfying ρ · ρ = 0, a problem which has been solved in the papers [6] and [33].…”

“…In this section we will prove a lemma that will bring together all the major players in corner scattering: complex geometrical optics solutions, the nonscattering wave, and the shape of the corner. This is the argument from [6,33]. We start by stating what we mean by a function having a specified order at a point:…”