Casimir energy due to inhomogeneous thin plates

Abstract

We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zerowidth, parallel mirrors with nonhomogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror’s plane. Using the Gelfand-Yaglom’s theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a derivative expansion, under the assumption that the spatial dependence of the properties is sufficiently smooth. We point out that, in some particular cases, the Casimir interaction energy for nonplanar perfect mirrors can be reproduced by inhomogeneities on planar mirrors.