The completeness of the generalized eigenfunctions and an upper bound for the counting function of the transmission eigenvalue problem for Maxwell equations

Abstract

Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.