We consider some lattices and look at discrete Laplacians on these lattices. In particular we look at solutions of the equation Δ(1)Φ = Δ(2)Z, where Δ(1) and Δ(2) denote two such Laplacians on the same lattice. We show that, in one dimension, when Δ(i), i = 1, 2, denote Δ(1)Φ = Φ(i + 1) + Φ(i - 1) - 2Φ(i) and Δ(2)Z = Z(i + 2) + Z(i - 2) - 2Z(i), this equation has a simple solution Φ(i) = Z(i + 1) + Z(i - 1) + 2Z(i). We show that in two dimensions, when the system is considered on a hexagonal (honeycomb) 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.
CITATION STYLE
Zakrzewski, W. J. (2005). Laplacians on lattices. Journal of Nonlinear Mathematical Physics, 12(4), 530–538. https://doi.org/10.2991/jnmp.2005.12.4.7
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