  In this paper we study the asymptotic behaviour as $\varepsilon\to 0$ of the
spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over
b^\varepsilon}\mathrm{div}(a^\varepsilon\nabla)$ posed in a bounded domain
$\Omega\subset\mathbb{R}^n$ $(n \geq 2)$ subject to Dirichlet boundary
conditions on $\partial\Omega$. When $\varepsilon\to 0$ both coefficients
$a^\varepsilon$ and $b^\varepsilon$ become high contrast in a small
neighborhood of a hyperplane $\Gamma$ intersecting $\Omega$. We prove that the
spectrum of $\mathcal{A}^\varepsilon$ converges to the spectrum of an operator
acting in $L^2(\Omega)\oplus L^2(\Gamma)$ and generated by the operation
$-\Delta$ in $\Omega\setminus\Gamma$, the Dirichlet boundary conditions on
$\partial\Omega$ and certain interface conditions on $\Gamma$ containing the
spectral parameter in a nonlinear manner. The eigenvalues of this operator may
accumulate at a finite point. Then we study the same problem, when $\Omega$ is
an infinite straight strip ("waveguide") and $\Gamma$ is parallel to its
boundary. We show that $\mathcal{A}^\varepsilon$ has at least one gap in the
spectrum when $\varepsilon$ is small enough and describe the asymptotic
behaviour of this gap as $\varepsilon\to 0$. The proofs are based on methods of
homogenization theory.
