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AbstractWe consider an eigenvalue problem for a divergence-form elliptic operator A?that has high-contrast periodic coefficients with period ?in each coordinate, where ?is a small parameter. The coefficients are perturbed on a bounded domain of ?order one? size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves?eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet?Bloch spectrum. For the so-called double porosity-type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in ?Then, using the tools of twoscale convergence for high-contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of A?. This implies that the eigenfunctions converge in the sense of strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator A0, consequently establishing ?asymptotic one-to-one correspondence? between the eigenvalues and the eigenfunctions of the operators A?and A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two-scale resolvent convergence of A?to A0but also the Hausdorff convergence of the spectra of A?to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues.

We consider an eigenvalue problem for a divergence‐form elliptic operator Aεthat has high‐contrast periodic coefficients with period εin each coordinate, where εis a small parameter. The coefficients are perturbed on a bounded domain of “order one” size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves—eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet–Bloch spectrum. For the so‐called double porosity‐type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in εThen, using the tools of twoscale convergence for high‐contrast homogenization, we prove the strong two‐scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of strong two‐scale convergence to the eigenfunctions of a two‐scale limit homogenized operator A0, consequently establishing “asymptotic one‐to‐one correspondence” between the eigenvalues and the eigenfunctions of the operators Aεand A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two‐scale resolvent convergence of Aεto A0but also the Hausdorff convergence of the spectra of Aεto the spectrum of A0, preserving the multiplicity of the isolated eigenvalues.