Asymptotics for minimal discrete energy on the sphere

Abstract:

We investigate the energy of arrangements of $N$ points on the surface of the unit sphere $S^d$ in $\mathbf {R}^{d+1}$ that interact through a power law potential $V = 1/r^s ,$ where $s > 0$ and $r$ is Euclidean distance. With $\mathcal {E}_d(s,N)$ denoting the minimal energy for such $N$-point arrangements we obtain bounds (valid for all $N$) for $\mathcal {E}_d(s,N)$ in the cases when $0 < s < d$ and $2 \leq d < s$. For $s = d$, we determine the precise asymptotic behavior of $\mathcal {E}_d(d,N)$ as $N \rightarrow \infty$. As a corollary, lower bounds are given for the separation of any pair of points in an $N$-point minimal energy configuration, when $s \geq d \geq 2$. For the unit sphere in $\mathbf {R}^3$ $(d = 2)$, we present two conjectures concerning the asymptotic expansion of $\mathcal {E}_2(s,N)$ 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).