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Singular Euler-Maclaurin expansion on multidimensional lattices
arXiv - CS - Numerical Analysis Pub Date : 2021-02-22 , DOI: arxiv-2102.10941 Andreas A. Buchheit, Torsten Keßler
arXiv - CS - Numerical Analysis Pub Date : 2021-02-22 , DOI: arxiv-2102.10941 Andreas A. Buchheit, Torsten Keßler
We extend the classical Euler-Maclaurin expansion to sums over
multidimensional lattices that involve functions with algebraic singularities.
This offers a tool for the precise quantification of the effect of microscopic
discreteness on macroscopic properties of a system. First, the Euler-Maclaurin
summation formula is generalised to lattices in higher dimensions, assuming a
sufficiently regular summand function. We then develop this new expansion
further and construct the singular Euler-Maclaurin (SEM) expansion in higher
dimensions, an extension of our previous work in one dimension, which remains
applicable and useful even if the summand function includes a singular function
factor. We connect our method to analytical number theory and show that all
operator coefficients can be efficiently computed from derivatives of the
Epstein zeta function. Finally we demonstrate the numerical performance of the
expansion and efficiently compute singular lattice sums in infinite
two-dimensional lattices, which are of high relevance in solid state and
quantum physics. An implementation in Mathematica is provided online along with
this article.
中文翻译:
多维格上的奇异Euler-Maclaurin展开
我们将经典的Euler-Maclaurin扩展扩展为涉及涉及代数奇异函数的多维晶格的求和。这提供了一种工具,用于精确量化微观离散对系统宏观特性的影响。首先,假设一个足够规则的求和函数,将欧拉-麦克劳伦求和公式推广到更高维度的晶格。然后,我们进一步开发此新扩展,并在更高的维度上构造奇异的Euler-Maclaurin(SEM)扩展,这是我们先前工作在一个维度上的扩展,即使summand函数包含奇异函数因子,它仍然适用且有用。我们将我们的方法与解析数论联系起来,并表明可以从Epstein zeta函数的导数有效地计算所有算子系数。最后,我们证明了展开的数值性能,并有效地计算了无限二维晶格中的奇异晶格和,这在固态和量子物理学中具有很高的相关性。与本文一起在线提供了Mathematica中的实现。
更新日期:2021-02-23
中文翻译:
多维格上的奇异Euler-Maclaurin展开
我们将经典的Euler-Maclaurin扩展扩展为涉及涉及代数奇异函数的多维晶格的求和。这提供了一种工具,用于精确量化微观离散对系统宏观特性的影响。首先,假设一个足够规则的求和函数,将欧拉-麦克劳伦求和公式推广到更高维度的晶格。然后,我们进一步开发此新扩展,并在更高的维度上构造奇异的Euler-Maclaurin(SEM)扩展,这是我们先前工作在一个维度上的扩展,即使summand函数包含奇异函数因子,它仍然适用且有用。我们将我们的方法与解析数论联系起来,并表明可以从Epstein zeta函数的导数有效地计算所有算子系数。最后,我们证明了展开的数值性能,并有效地计算了无限二维晶格中的奇异晶格和,这在固态和量子物理学中具有很高的相关性。与本文一起在线提供了Mathematica中的实现。