Transmission Eigenvalues for a Class of Non-Compactly Supported Potentials

Abstract

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

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