A Note on Classical Ground State Energies

The pair-specific ground state energy ε _g (N):=ℰ _g (N)/(N(N−1)) of Newtonian N body systems grows monotonically in N. This furnishes a whole family of simple new tests for minimality of putative ground state energies ℰ _g ^x (N) obtained through computer experiments. Inspection of several publicly available lists of such computer-experimentally obtained putative ground state energies ℰ _g ^x (N) has yielded several dozen instances of ℰ _g ^x (N) which failed one of these tests; i.e., for those N one concludes that ℰ _g ^x (N)>ℰ _g (N) strictly. Although the correct ℰ _g (N) is not revealed by this method, it does yield a better upper bound on ℰ _g (N) than ℰ _g ^x (N) whenever ℰ _g ^x (N) fails a monotonicity test. The surveyed N-body systems include in particular N point charges with 2- or 3-dimensional Coulomb pair interactions, placed either on the unit 2-sphere or on a 2-torus (a.k.a. Thomson, Fekete, or Riesz problems).