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

TL;DR

This paper refines the understanding of the van der Waals interaction energy at large distances, providing sharper estimates on the remainder and an improved upper bound under weaker assumptions for spinless Fermions.

Contribution

It offers new remainder estimates and a sharper upper bound for the van der Waals interaction energy, simplifying the proof and weakening the assumptions compared to previous results.

Findings

Established sharper remainder estimates for the interaction energy.

Proved an upper bound that is sharp to leading order.

Improved upon Lieb and Thirring's upper bound under weaker assumptions.

Abstract

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.