Volume 12, Issue 4, November 2005, Pages 530 - 538
Laplacians on Lattices
Authors
Wojtek J. Zakrzewski
Corresponding Author
Wojtek J. Zakrzewski
Received 27 January 2005, Accepted 7 March 2005, Available Online 1 November 2005.
- DOI
- 10.2991/jnmp.2005.12.4.7How to use a DOI?
- 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
- © 2006, the Authors. Published by Atlantis Press.
- Open Access
- This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Cite this article
TY - JOUR AU - Wojtek J. Zakrzewski PY - 2005 DA - 2005/11/01 TI - Laplacians on Lattices JO - Journal of Nonlinear Mathematical Physics SP - 530 EP - 538 VL - 12 IS - 4 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2005.12.4.7 DO - 10.2991/jnmp.2005.12.4.7 ID - Zakrzewski2005 ER -