SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1474-1491, January 2021.
The electrostatic potentials $\phi$ associated with neutral periodic crystals are defined by lattice sums that are never absolutely convergent. The sum depends on the order of summation. The mean zero periodic solution of Poisson's equation, denoted ${\underline \phi}$, is a natural potential. So is the potential obtained by the Mellin transform algorithm of [Borwein, Borwein, and Taylor, J. Math. Phys., 26 (1985), pp. 2999--3009]. We prove that these two are equal and are both equal to the potential obtained by Abel summation. The sum defining $\partial^\alpha \phi$ converges absolutely for $|\alpha|\ge 3$ to $\partial^\alpha{\underline \phi}$. The indeterminacy in the potential is at most a harmonic polynomial of degree 2.

中文翻译：

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周期性晶体的静电势

SIAM数学分析杂志，第53卷，第2期，第1474-1491页，2021
年1月。与中性周期晶体相关的静电势$ \ phi $由从未绝对收敛的晶格和定义。总和取决于求和的顺序。泊松方程的平均零周期解表示为$ {\下划线\ phi} $，是自然势。通过[Borwein，Borwein和Taylor，J. Math。Phys。，26（1985），第2999--3009页]。我们证明这两个是相等的，并且都等于通过Abel求和获得的势。定义$ \ partial ^ \ alpha \ phi $的总和对于$ | \ alpha | \ ge 3 $绝对收敛到$ \ partial ^ \ alpha {\ underline \ phi} $。电位的不确定性最多为2阶的谐波多项式。