Periodic Riesz and Gauss-Type Potentials

Abstract

In this chapter, we consider kernels of the form \(K(x, y):=F(x-y)\) for some F periodic with respect to a given lattice \(\varLambda \subset \mathbb {R}^d\); that is, \(F(x+v)=F(x)\) for all \(v\in \varLambda \). In Section 10.1, we consider periodic F defined by lattice sums of the form \(F(x)=\sum _{v\in \varLambda }f(x+v), \) for some lattice \(\varLambda \) and f with sufficient decay (see Definition 10.1.1) for the sum to converge absolutely (or to \(\infty \) unconditionally). Such sums represent the energy required to place a unit charge at x in the presence of unit charges located on \(\varLambda \) with pairwise interactions given by f. In this section, we also define and relate several notions of energy related to these periodic potentials.