Journal of Nonlinear Mathematical Physics

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
https://doi.org/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.
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
12 - 4
Pages
530 - 538
Publication Date
2005/11
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.2005.12.4.7How to use a DOI?
Open Access
This is an open access article distributed under the CC BY-NC license.

Cite this article

TY  - JOUR
AU  - Wojtek J ZAKRZEWSKI
PY  - 2005
DA  - 2005/11
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  - https://doi.org/10.2991/jnmp.2005.12.4.7
ID  - ZAKRZEWSKI2005
ER  -