Moment methods in energy minimization: New bounds for Riesz minimal \n energy problems

Abstract

We use moment methods to construct a converging hierarchy of optimization \nproblems to lower bound the ground state energy of interacting particle \nsystems. We approximate the infinite dimensional optimization problems in this \nhierarchy by block diagonal semidefinite programs. For this we develop the \nnecessary harmonic analysis for spaces consisting of subsets of another space, \nand we develop symmetric sum-of-squares techniques. We compute the second step \nof our hierarchy for Riesz $s$-energy problems with five particles on the \n$2$-dimensional unit sphere, where the $s=1$ case known as the Thomson problem. \nThis yields new sharp bounds (up to high precision) and suggests the second \nstep of our hierarchy may be sharp throughout a phase transition and may be \nuniversally sharp for $5$-particles on $S^2$. This is the first time a \n$4$-point bound has been computed for a continuous problem