Transmission Eigenvalues for a Class of Non-Compactly Supported Potentials

Let $Ω\subseteq\mathbb R^n$ be a non-empty open set for which the Sobolev embedding $H_0^2(Ω)\longrightarrow L^2(Ω)$ is compact, and let $V\in L^\infty(Ω)$ be a potential taking only positive real values and satisfying the asymptotics $V(\cdot)\asymp\left\langle\cdot\right\rangle^{-α}$ for some $α\in\left]3,\infty\right[$. We establish the discreteness of the set of real transmission eigenvalues for both Schrödinger and Helmholtz scattering with these potentials.

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