Spectral structures of elastic-electromagnetic transmission eigenvalue problems

Abstract The time-harmonic elastic-electromagnetic interior transmission eigenvalue problem (EEITEP) arises when an elastic body becomes invisible to an incident electromagnetic wave. This spectral problem is typically non-elliptic and non-self-adjoint, making its analysis delicate. In this paper, we study the discreteness of transmission eigenvalues and the boundary localization of the associated eigenfunctions. For a general bounded Lipschitz domain, we prove that the set of positive transmission eigenvalues, if non-empty, is discrete with $\infty$ as its only possible accumulation point. For a radially symmetric domain, we demonstrate the existence of a sequence of transmission eigenvalues and derive their asymptotic behavior. We rigorously show that the corresponding transmission eigenfunctions exhibit boundary localization in their electromagnetic components, whereas the elastic displacement field remains globally distributed throughout the domain. Finally, we derive lower bounds for the $L^{\infty}$-norms of the electromagnetic gradients normalized by their $L^2$-norms, quantifying their blow-up behavior near the boundary along this sequence. These findings reveal a potential spectral mechanism for developing super-resolution imaging methods in elastic-electromagnetic scattering.