Asymptotic expansion of transmission eigenvalues for anisotropic thin layers

We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second-order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [Nonlinear eigenvalue approximation for compact operators. J Math Phys. 2015;56(11):113512]. We also provide formal derivation in more general cases validating the obtained theoretical result.