ASYMPTOTIC THEORY FOR A GENERAL CLASS OF SHORT-RANGE INTERACTION FUNCTIONALS

In models of N interacting particles in R-d as in Density Functional Theory or crowd motion, the repulsive cost is usually described by a two-point function c(epsilon)(x,y)=& ell;(|x-y|/epsilon) where & ell;:R+->[0,infinity] is decreasing to zero at infinity and parameter epsilon>0 scales the interaction distance. In this paper we identify the limit energy of such a model in the short-range regime epsilon << 1 under the sole assumption that there exists r(0)>0 : integral r(0)(infinity)& ell;(r)r(d-1)dr<+infinity. This extends recent results [D. Hardin, E. B. Saff, and O. Vlasiuk, Asymptotic Properties of Short-Range Interaction Functionals, preprint, https://arxiv.org/abs/2010.11937, 2021], [D. P. Hardin, T. Lebl & eacute;, E. B. Saff, and S. Serfaty, Constr. Approx., 48 (2018), pp. 61-100], [M. Lewin, J. Math. Phys., 63 (2022), 061101] obtained in the homogeneous case & ell;(r)=r(-s) where s>d.