Neutral point sources and Fourier data on a three-dimensional torus

Let $A\in G{L}_3(\mathbb{R})$, let $\mathrm{\Lambda }=A{\mathbb{Z}}^3$, let $V=\mathrm{\mid }\det A\mathrm{\mid }$, and let ${\mathbb{T}}_A={\mathbb{R}}^3\mathrm{/}\mathrm{\Lambda }$. These are the full-rank lattice and covolume conventions of Full-rank lattices, covolume, and the dual lattice. The reciprocal lattice is ${\mathrm{\Lambda }}^{\ast }=A^{-T}{\mathbb{Z}}^3$. For $k\in {\mathbb{Z}}^3$ write ${\kappa }_k=2\pi A^{-T}k,e_k(x)=e^{i{\kappa }_k\cdot x}\mathrm{.}$ Thus $e_k$ is a character of ${\mathbb{T}}_A$; the ordinary Fourier frequency is $A^{-T}k$, while ${\kappa }_k$ is the angular physical frequency.

A finite point-source configuration consists of an integer $n\ge 0$, points $x_1,\dots ,x_n\in {\mathbb{T}}_A$ that are pairwise distinct when $n>1$, and real charges $q_1,\dots ,q_n$. It is neutral when ${\sum }_{i=1}^n{q}_i=0$. For a smooth periodic test function $\phi$, the point-source distribution is the linear functional $⟨\rho ,\phi ⟩={\sum }_{i=1}^n{q}_i\phi (x_i)\mathrm{.}$ Here smooth means that the periodic lift to ${\mathbb{R}}^3$ has continuous derivatives of every order, and the test-function topology is generated by the seminorms ${\max }_{x\in A[0,1{]}^3}\mathrm{\mid }{\mathrm{\partial }}^{\beta }\phi (x)\mathrm{\mid }$ over all derivative multi-indices $\beta$. A periodic distribution is a continuous complex-linear functional for this topology. Its distributional Laplacian is defined by $⟨-\mathrm{\Delta }u,\phi ⟩=⟨u,-\mathrm{\Delta }\phi ⟩$.

For a smooth periodic $\phi$, its Fourier coefficient is $\widehat{\phi }(k)=\frac{1}{V}{\int }_{A(0,1{]}^3}\phi (x)e^{-i{\kappa }_k\cdot x}dx\mathrm{.}$ The source coefficient is $\widehat{\rho }(k)=\frac{1}{V}{\sum }_{i=1}^n{q}_i{e}^{-i{\kappa }_k\cdot x_i};$ in particular, neutral data have $\widehat{\rho }(0)=0$. For a periodic distribution $u$, define $\widehat{u}(k)=V^{-1}⟨u,e_{-k}⟩$. The mean-zero periodic Poisson solution is a periodic distribution $u$ satisfying $-\mathrm{\Delta }u=4\pi \rho$ and $\widehat{u}(0)=0$. This definition fixes the zero-mode convention; no nonneutral periodic solution is implicit.

The Coulomb energy convention is the half-sum of each source charge times the regular part of its potential, after subtracting its own singular term $q_i\mathrm{/}\mathrm{\mid }x-x_i\mathrm{\mid }$. The resulting finite part is called the renormalized self-energy convention. Its Ewald formula is established in this development.

• The Coulomb normalization is $-\mathrm{\Delta }u=4\pi \rho$ in three dimensions. No dielectric factor, surface term, nonneutralizing background, or alternative boundary condition is part of this definition.
• The dual lattice in the Poisson formula uses exponent $2\pi i\xi \cdot x$; multiplying its frequency $\xi =A^{-T}k$ by $2\pi$ gives ${\kappa }_k$.