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
DOI:
https://doi.org/10.1090/S0002-9947-98-02119-9
MathSciNet review:
1458327
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We investigate the energy of arrangements of points on the surface of the unit sphere
in
that interact through a power law potential
where
and
is Euclidean distance. With
denoting the minimal energy for such
-point arrangements we obtain bounds (valid for all
) for
in the cases when
and
. For
, we determine the precise asymptotic behavior of
as
. As a corollary, lower bounds are given for the separation of any pair of points in an
-point minimal energy configuration, when
. For the unit sphere in
, we present two conjectures concerning the asymptotic expansion of
that relate to the zeta function
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
when
(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
Email:
maarno@math.cityu.edu.hk
E. B. Saff
Affiliation:
Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email:
esaff@math.usf.edu
DOI:
https://doi.org/10.1090/S0002-9947-98-02119-9
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