Atlantis Press Paper Details: Laplacians on Lattices

We consider some lattices and look at discrete Laplacians on these lattices. In partiular we look at solutions of the equation (1) = (2)Z, where (1) and (2) denote two such Laplacians on the same lattice. We show that, in one dimension, when (i), i = 1, 2, denote (1) = (i + 1) + (i - 1) - 2(i) and (2)Z = Z(i + 2) + Z(i - 2) - 2Z(i), this equation has a simple solution (i) = Z(i + 1) + Z(i - 1) + 2Z(i). We show that in two dimensions, when the system is considered on a hexagonal (hoeycomb) lattice, we have a similar relation. This is also true in three dimensions when we have a very special lattice (tetrahedral with points inside). We also briefly discuss how this relation generalizes when we consider other lattices.


title:		Laplacians on Lattices
publication:		JNMP
volume-issue:		12 - 4
pages:		530 - 538
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.2005.12.4.7 (how to use a DOI)
author(s):		Wojtek J ZAKRZEWSKI
publication date:		November 2005
abstract:		We consider some lattices and look at discrete Laplacians on these lattices. In partiular we look at solutions of the equation (1) = (2)Z, where (1) and (2) denote two such Laplacians on the same lattice. We show that, in one dimension, when (i), i = 1, 2, denote (1) = (i + 1) + (i - 1) - 2(i) and (2)Z = Z(i + 2) + Z(i - 2) - 2Z(i), this equation has a simple solution (i) = Z(i + 1) + Z(i - 1) + 2Z(i). We show that in two dimensions, when the system is considered on a hexagonal (hoeycomb) lattice, we have a similar relation. This is also true in three dimensions when we have a very special lattice (tetrahedral with points inside). We also briefly discuss how this relation generalizes when we consider other lattices.
copyright:		© Atlantis Press. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
full text:		124art4.pdf (91 K)