Casimir energy of the quantum field in a dispersive and absorptive medium

We calculate the Casimir energy for a quantum field between two metallic plates separated by a dielectric in the microscopic framework with the medium modeled by a set of oscillators with continuously distributed frequencies. We study the case of one spatial dimension and compare the results for the additional energy and force due to interaction with the medium with the results obtained in the framework of the phenomenological approach similar to the well-known Lifshitz formula: they have opposite signs.