In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, but defined so that it has meaning on a graph or a discrete grid.

Definition

Let G=(V,E) be a graph with vertices V and edges E. Let be a function mapping vertices to complex numbers. Then, the discrete Laplacian Δ acting on φ is defined by

(Δφ)(v) =	∑	φ(w) - φ(v)
	w:dist(w,v) = 1

where dist(w,v) is the distance operator on the graph. Thus, this sum is over the nearest neighbors of the vertex v.

If the graph has weighted edges, that is, a weighting function is given, then the definition can be generalized to

(Δγφ)(v) =	∑	γwv(φ(w) - φ(v))
	w:dist(w,v) = 1

where γ_wv is the weight value on the edge .

Theorems

If the graph is a square lattice grid, then this definition of the Laplacian can be shown to correspond to the continuous Laplacian in the limit of an infinitely fine grid. Thus, for example, on a one-dimensional grid we have

This definition of the Laplacian is commonly used in numerical analysis and in image processing. In image processing, it is considered to be a type of digital filter, more specifically an edge filter, called the Laplace filter .

Discrete Schrödinger operator

Let be a potential function defined on the graph. Note that P can be considered to be a multiplicative operator acting diagonally on φ

(Pφ)(v) = P(v)φ(v).

Then H = Δ + P is the discrete Schrödinger operator, an analog of the continuous Schrödinger operator.

If the number of edges meeting at a vertex is uniformly bounded, and the potential is bounded, then H is bounded and self-adjoint.

The spectral properties of this Hamiltonian can be studied with Stone's theorem; this is a consequence of the duality between posets and boolean algebras.

Discrete Green's function

The Green's function of the discrete Schrödinger operator is given in the resolvent formalism by

where δ_w is understood to be the Kronecker delta function on the graph: δ_w(v) = δ_wv; that is, it equals 1 if v=w and 0 otherwise.

For fixed and λ a complex number, the Green's function considered to be a function of v is the unique solution to

(H - λ)G(v,w;λ) = δ_w(v).