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 $S^2\\subset\\mathbb{R}^3$ that gives an equal weight cubature rule (or equal weight numerical integration rule) on $S^2$ which is exact for spherical polynomials of degree $\\leqslant n$. (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 $X\\in\\Xi$ the minimum spherical distance between points is bounded from below by $\\lambda/\\sqrt{m}$.) 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 $S^2$.