The Coulomb energy of spherical designs on S ²

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 ℝ³ 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 Ξ of m-point spherical n-designs X on S ² is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X Ξ the minimum spherical distance between points is bounded from below by λ 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.