  We study configurations of points on the unit sphere that minimize potential
energy for a broad class of potential functions (viewed as functions of the
squared Euclidean distance between points). Call a configuration sharp if there
are m distances between distinct points in it and it is a spherical
(2m-1)-design. We prove that every sharp configuration minimizes potential
energy for all completely monotonic potential functions. Examples include the
minimal vectors of the E_8 and Leech lattices. We also prove the same result
for the vertices of the 600-cell, which do not form a sharp configuration. For
most known cases, we prove that they are the unique global minima for energy,
as long as the potential function is strictly completely monotonic. For certain
potential functions, some of these configurations were previously analyzed by
Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize
our results to other compact two-point homogeneous spaces, and we conclude with
an extension to Euclidean space.
