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Remainder estimates for the Long Range Behavior of the van der Waals interaction energy

The van der Waals-London's law, for a collection of atoms at large separation, states that their interaction energy is pairwise attractive and decays proportionally to one over their distance to the sixth. The first rigorous result in this direction was obtained by Lieb and Thirring [LT], by proving an upper bound which confirms this law. Recently the van der Waals-London's law was proven under some assumptions by I.M. Sigal and the author [AS]. Following the strategy of [AS] and reworking the approach appropriately, we prove estimates on the remainder of the interaction energy. Furthermore, using an appropriate test function, we prove an upper bound for the interaction energy, which is sharp to leading order. For the upper bound, our assumptions are weaker, the remainder estimates stronger and the proof is simpler. The upper bound, for the cases it applies, improves considerably the upper bound of Lieb and Thirring. However, their bound is much more general. Here we consider only spinless Fermions.