On the convergence of bloch eigenfunctions in homogenization problems

We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. Two cases are considered, the case of classical homogenization, where the coefficient matrix satisfies the ellipticity condition uniformly with respect to $\varepsilon$, and the case of two-scale homogenization, where the coefficient matrix has two phases and is highly contrast with hard-to-soft-phase contrast ratio $1\,{:}\,\varepsilon^2$.