Uniform spectral asymptotics for singularly perturbed locally periodic operators

We consider the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficients are scaled as ε² and vary both on the macroscopic scale and on the periodic microscopic scale. We make a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior. We then prove an exponential localization phenomena at this minimum point. Namely, the k-th original eigenfunction is shown to be asymptotically given by the product of the first cell eigenfunction (at the ε scale) times the k-th eigenfunction of an homogenized problem (at the √ε scale). The homogenized problem is a diffusion equation with quadratic potential in the whole space. We first perform asymptotic expansions, and then prove convergence by using a factorization strategy.