  A classification of the electromagnetic modes on open and closed spatial
domains containing media with piecewise homogeneous permittivities is used to
facilitate the derivation of quantum induced Casimir stresses in dielectrics.
By directly exploiting the complex analytic properties of solutions of the
macroscopic Maxwell equations for open systems it is shown how regular
expressions for such stresses can be expressed in terms of double integrals
involving either real or pure imaginary frequencies associated with harmonic
modes in conformity with the Lifshitz theory for separated planar dielectric
half-spaces. The derivation is self-contained without recourse to the Krein
formula for a density of states or mode regularization and offers a more direct
approach to other open systems.
