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Finite size effects and non-additivity in the van der Waals interaction

We obtain analytically the exact non-retarded dispersive interaction energy between an atom and a perfectly conducting disc. We consider the atom in the symmetry axis of the disc and assume the atom is predominantly polarizable in the direction of this axis. For this situation we discuss the finite size effects on the corresponding interaction energy. We follow the recent procedure introduced by Eberlein and Zietal together with the old and powerful Sommerfeld's image method for non-trivial geometries. For the sake of clarity we present a detailed discussion of Sommerfeld's image method. Comparing our results form the atom-disc system with those recently obtained for an atom near a conducting plane with a circular aperture, we discus the non-additivity of the van der Waals interactions involving an atom and two complementary surfaces. We show that there is a given ratio z/a between the distance z from the atom to the center of the disc (aperture) and the radius of the disc a (aperture) for which non-additivity effects vanish. Qualitative arguments suggest that this quite unexpected result will occur not only for a circular hole, but for anyother symmetric hole.