Energy minimization, periodic sets and spherical designs

Renaud Coulangeon (1) , Achill Schürmann (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Institut für Mathematik

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.

Domaines: Géométrie métrique [math.MG]; Théorie des nombres [math.NT]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Renaud Coulangeon, Achill Schürmann. Energy minimization, periodic sets and spherical designs. International Mathematics Research Notices, 2012, 4, pp.829-848. ⟨10.1093/imrn/rnr048⟩. ⟨hal-00775916⟩