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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. [ABSTRACT FROM AUTHOR]

We consider the transmission eigenvalue problem for a medium surrounded by a thin layer of inhomogeneous material with different refractive index. We derive explicit asymptotic expansion for the transmission eigenvalues with respect to the thickness of the thin layer. We prove the error estimate for the asymptotic expansion up to order 1. This expansion can be used to obtain explicit expressions for a constant index of refraction. [ABSTRACT FROM AUTHOR]