Transmission eigenvalues for a class of non-compactly supported potentials

Abstract

Let be a non-empty open set for which the Sobolev embedding is compact, and let V ∈ L∞(Ω) be a potential taking only positive real values and satisfying the asymptotics V( · )≈〈 · 〉−α for some α ∈ ]3, ∞[. We establish the discreteness of the set of real transmission eigenvalues for both Schrödinger and Helmholtz scattering with these potentials.