van der Waals energy and pressure in dissipative media: Fluctuational electrodynamics and mode summation

We derive the van der Waals energy and pressure in a planar multilayer system with arbitrary number of dissipative films between two half-spaces. A unique feature of this work is that the entire analysis is performed on the real frequency axis instead of summation over Matsubara frequencies on the imaginary frequency axis. The expression we obtain for van der Waals energy is a generalization of van Kampen and Schram's result for dissipationless media. By considering a specific case of a vacuum gap between multilayer objects with dissipative films, we show that the van der Waals energy due to the vacuum gap can not be interpreted simply as a sum of free energy of normal modes.