Spectral convergence for high contrast media with a defect via homogenization

We consider an eigenvalue problem for a divergence form elliptic operator Aε with locally perturbed high contrast periodic coefficients. Periodicity size ε is a small parameter. Local perturbation of coefficients for such operator could result in emergence of localized waves eigenfunctions with corresponding eigenvalues lying in the gaps of the Floquet-Bloch spectrum. We prove that, for a double porosity type scaling, under some natural assumptions the eigenfunctions decay exponentially at infinity, uniformly in ε. Then, using the tools of two-scale convergence for high contrast homogenization, we prove the two-scale compactness of the normalized eigenfunctions of Aε, i.e. that, up to a subsequence, they two-scale converge to the eigenfunctions of a two-scale limit homogenized operator A0. This consequently establishes asymptotic one-to-one correspondence between eigenvalues and eigenfunctions of Aε and A0. We also prove by direct means the stability of the essential spectrum of the homogenized operator with respect to the local perturbation of its coefficients. That allows us to establish not only strong two-scale resolvent convergence of Aε to A0 but also Hausdorff convergence of the spectra of Aε to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues.