Spectral properties of operators and wave propagation in high-contrast media

The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains with Dirichlet or Neumann boundary conditions, as well as in infinite periodic media, is proposed. For a small parameter $\epsilon >0$ characterizing the contrast of the components of the medium, the analyticity of the eigenvalues and eigenfunctions is established in a neighborhood of $\epsilon =0$. Effective operators corresponding to $\epsilon =0$ are described.