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# Discrete Laplace operator

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.

 Contents

## Definition

Let G=(V,E) be a graph with vertices V and edges E. Let $\phi:V\rightarrow\mathbb{C}$ 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 $\gamma:E\rightarrow\mathbb{C}$ is given, then the definition can be generalized to

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

where γwv is the weight value on the edge $wv\in E$.

## 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

$\frac{\partial^2F}{\partial x^2} = \lim_{\epsilon \rightarrow 0} \frac{(F(x+\epsilon)-F(x))+(F(x-\epsilon)-F(x))}{\epsilon^2}.$

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 $P:V\rightarrow\mathbb{R}$ 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

$G(v,w;\lambda)=\langle\delta_v| \frac{1}{H-\lambda}| \delta_w\rangle$

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

For fixed $w\in V$ 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) = δw(v).
Last updated: 10-12-2005 11:53:39
03-10-2013 05:06:04
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