Lattice Green's functions in all dimensions
Lattice Green's functions in all dimensions
复制标题
所有维度的格格林函数
DOI：
10.1088/1751-8113/43/30/305205
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
A. Guttmann
中科院分区：
文献类型：
--
作者：
A. Guttmann
本文系统地讨论了d维金刚石、简单立方、体心立方和面心立方晶格上的格点绿色函数（LGF），其中前三种晶格的格点格林函数为任意维数d <$2，超面心立方晶格的格点格林函数为2 <$d <$5。我们证明了d维超立方晶格的LGF和（d − 1）维金刚石晶格的LGF之间有着密切的联系。我们给这些晶格中的每一个在所有维度的LGF的常数项配方。通过与马勒测度的联系，我们指出了sc，bcc和金刚石LGF的系数与1/π的一些Ramanujan型公式之间的意想不到的联系。
We give a systematic treatment of lattice Green's functions (LGF) on the d-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality d ⩾ 2 for the first three lattices, and for 2 ⩽ d ⩽ 5 for the hyper-fcc lattice. We show that there is a close connection between the LGF of the d-dimensional hyper-cubic lattice and that of the (d − 1)-dimensional diamond lattice. We give constant-term formulations of LGFs for each of these lattices in all dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the sc, bcc and diamond LGFs and some Ramanujan-type formulae for 1/π.