Transactions of the American Mathematical Society vol:350 issue:2 pages:523-538

Abstract:

We investigate the energy of arrangements of N points on the surface of the unit sphere S^d in R^(d+1) that interact through a power law potential V = 1/r^s, where s > 0 and r is Euclidean distance. With E_d(s, N) denoting the minimal energy for such N-point arrangements we obtain bounds (valid for all N) for E_d(s, N) in the cases when 0 < s < d and 2 <= d < s. For s = d, we determine the precise asymptotic behavior of E_d(d, N) as N -> infinity. As a corollary, lower bounds are given for the separation of any pair of points in an N-point minimal energy configuration, when s >= d >= 2. For the unit sphere in R^3 (d = 2), we present two conjectures concerning the asymptotic expansion of E_2(s, N) that relate to the zeta function zeta_L(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of zeta_L)(s) when 0 < s < 2 (the divergent case).