Next Order Energy Asymptotics for Riesz Potentials on Flat Tori

作者：Hardin Douglas P^*; Saff Edward B; Simanek Brian Z; Su Yujian

来源：International Mathematics Research Notices, 2017, (12): 3529-3556.

DOI：10.1093/imrn/rnw049

Let Lambda be a lattice in R-d with positive co-volume. Among Lambda-periodic N-point configurations, we consider the minimal renormalized Riesz s-energy epsilon(s,Lambda) (N). While the dominant term in the asymptotic expansion of epsilon(s,Lambda) (N) as N goes to infinity in the long-range case that 0 < s < d (or s = log) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for s > 0 they are of the form C-s,C-d|Lambda|-N-s/d(1+s/d) and -2/dNlogN + (C-log,C-d -2 zeta'(Lambda)(0)) N where we show that the constants C-s,C-d and C-log,C-d are independent of the lattice Lambda.

• 出版日期2017-6