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We consider an eigenvalue problem for a divergence form elliptic operator$A_\epsilon$ with high contrast periodic coefficients with period $\epsilon$ ineach coordinate, where $\epsilon$ is a small parameter. The coefficients areperturbed on a bounded domain of `order one' size. The local perturbation ofcoefficients for such operator could result in emergence of localized waves -eigenfunctions with corresponding eigenvalues lying in the gaps of theFloquet-Bloch spectrum. We prove that, for the so-called double porosity typescaling, the eigenfunctions decay exponentially at infinity, uniformly in$\epsilon$. Then, using the tools of two-scale convergence for high contrasthomogenization, we prove the strong two-scale compactness of the eigenfunctionsof $A_\epsilon$. This implies that the eigenfunctions converge in the sense ofthe strong two-scale convergence to the eigenfunctions of a two-scale limithomogenized operator $A_0$, consequently establishing `asymptotic one-to-onecorrespondence' between the eigenvalues and the eigenfunctions of these twooperators. We also prove by direct means the stability of the essentialspectrum of the homogenized operator with respect to the local perturbation ofits coefficients. That allows us to establish not only the strong two-scaleresolvent convergence of $A_\epsilon$ to $A_0$ but also the Hausdorffconvergence of the spectra of $A_\epsilon$ to the spectrum of $A_0$, preservingthe multiplicity of the isolated eigenvalues.

Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization