  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
$\varepsilon$ tends to zero, the eigenvalues are of order $\varepsilon^{-2}$
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 $\sqrt{\varepsilon}$.
