Geometric Weakening of the Casimir Interaction

We examine the dependence of the Casimir interaction between separate (metallic) bodies on their geometry. From a semi-classical point of view it depends strongly on whether the dominant periodic orbits are stable or unstable and on the number of focal points. We give a very simple semiclassical argument for the theorem [1] that mirror-symmetric periodically corrugated metallic surfaces always attract. Although counter-intuitive because the Van DerWaals interactions between individual pairs of atoms are attractive at long range, we argue that this need not be the case for multi-atom interactions. Semi-classical methods are used to determine the shape of surfaces with minimal Casimir interaction. [1] O.Kenneth, I. Klich, Phys.Rev.Lett. 97, 160401 (2006)~