The Coulomb energy of spherical designs on S2

In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n- designs, where a spherical n- design is a set of m points on the unit sphere S-2 subset of R-3 that gives an equal weight cubature rule ( or equal weight numerical integration rule) on S-2 which is exact for spherical polynomials of degree <= n. ( A sequence Xi of m- point spherical n- designs X on S-2 is said to be well separated if there exists a constant lambda>0 such that for each m- point spherical n- design X epsilon Xi the minimum spherical distance between points is bounded from below by. lambda/root m.) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n(2)), then the Coulomb energy of each m- point spherical n- design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S-2.