Localization of eigenfunctions in a thin domain with locally periodic oscillating boundary

We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin domain. The boundary of the domain is assumed to be locally periodic. When the thickness of the domain $\epsilon$ tends to zero, the eigenvalues are of order ${\epsilon }^{-2}$ and described in terms of the first eigenvalue $\mu (x_1)$ of an auxiliary spectral cell problem parametrized by $x_1$, while the eigenfunctions localize with rate $\sqrt{\epsilon }$.