Laplacians on lattices

Abstract

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.