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.