Transmission Eigenvalues for a Class of Non-Compactly Supported Potentials

TL;DR

This paper proves the discreteness of real transmission eigenvalues for Schrödinger and Helmholtz scattering problems involving certain non-compactly supported potentials with specific asymptotic decay.

Contribution

It establishes the discreteness of transmission eigenvalues for a class of potentials that decay at infinity, extending previous results to non-compactly supported cases.

Findings

Transmission eigenvalues are discrete for the considered potentials.

Results apply to both Schrödinger and Helmholtz scattering.

Potential decay rate influences eigenvalue properties.

Abstract

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