Laplacians on Lattices

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/).

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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  -