Mathematical Research Letters
Volume 1, Issue 6, November 1994 pp. 647-662.
Minimal Discrete Energy on the SphereAuthors: E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou
Author institution: Steklov Institute, and University of South Florida
Summary: We investigate the energy of arrangements of $N$ points on the surface of a sphere in ${\bold R}^3$, interacting through a power law potential $V=r^\alpha$, $-2 < \alpha < 2$, where $r$ is Euclidean distance. For $\alpha=0$, we take $V=\log(1/r)$. An area-regular partitioning scheme of the sphere is devised for the purpose of obtaining bounds for the extremal (equilibrium) energy for such points. For $\alpha=0$, finer estimates are obtained for the dominant terms in the minimal energy by considering stereographical projections on the plane and analyzing certain logarithmic potentials. A general conjecture on the asymptotic form (as $N \to \infty$) of the extremal energy, along with its supporting numerical evidence, is presented. Also we introduce explicit sets of points, called ``generalized spiral points", that yield good estimates for the extremal energy. At least for $N \leq 12,\,000$ these points provide a reasonable solution to a problem of M. Shub and S. Smale arising in complexity theory.
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