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

TL;DR

This paper proves the completeness of generalized eigenfunctions and provides an optimal upper bound for the transmission eigenvalue counting function for Maxwell equations, under certain regularity conditions.

Contribution

It establishes the completeness of eigenfunctions and derives an optimal upper bound for the eigenvalue counting function using spectral theory, extending previous results.

Findings

Completeness of generalized eigenfunctions is proven.

An optimal upper bound for the eigenvalue counting function is derived.

Results apply under twice continuously differentiable coefficients.

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.