Spectral gap properties of perturbed periodic media

We analyze periodic operators on Rn and small perturbations of these operators. The perturbation is periodic in n − 1 directions and has bounded support in the remaining direction. We show that, when the perturbation has a sign, every spectral gap for the unperturbed operator is reduced by the perturbation. We develop a general theory that can be applied to elliptic operators, to systems such as that of linear elasticity, and to Maxwell’s equations.