An investigation of the convergence of Casimir stress in one-dimensional inhomogeneous media

This paper, based on the Lifshitz theory and the existing regularization employed to calculate Casimir stress, does further mathematics deduction on them to explore the convergence of Casimir stress in one dimension. By analyzing the Galerkin variational equation of Green's function of electromagnetic field, this paper strictly proves and concludes that under the existing Lifshitz formulism and standard regularization, as long as either the derivative of permittivity ε or the derivative of permeability μ is nonzero somewhere in the media, Casimir stress is divergent in this place with nonzero derivative. This investigation proves that existing standard regularization are not applicable to inhomogeneous media, which provides theoretical reference for the improvement of the current physical model and the exploration of new regularization applicable to inhomogeneous media.