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Transactions of the American Mathematical Society

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Asymptotics for minimal discrete energy on the sphere

Authors: A. B. J. Kuijlaars and E. B. Saff
Journal: Trans. Amer. Math. Soc. 350 (1998), 523-538
MSC (1991): Primary 52A40; Secondary 31C20, 41A60
MathSciNet review: 1458327
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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).

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Additional Information

A. B. J. Kuijlaars
Affiliation: Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Address at time of publication: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
MR Author ID: 341696

E. B. Saff
Affiliation: Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Tampa, Florida 33620
MR Author ID: 152845

Received by editor(s): October 9, 1995
Additional Notes: The first author is supported by the Netherlands Foundation for Mathematics SMC with financial aid from the Netherlands Organization for the Advancement of Scientific Research (NWO). This research was done while visiting the University of South Florida, Tampa
The research of the second author is supported, in part, by the U.S. National Science Foundation under grant DMS-9501130.
Article copyright: © Copyright 1998 American Mathematical Society