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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 <math>\phi:V\rightarrow\mathbb{C}<math> be a function mapping vertices to complex numbers. Then, the discrete Laplacian <math>\Delta<math> acting on <math>\phi<math> is defined by
- <math>(\Delta\phi)(v)=\sum_{w:\textrm{dist}(w,v)=1}\phi(w)-\phi(v)<math>
where <math> \textrm{dist}(w,v)<math> 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 <math>\gamma:E\rightarrow\mathbb{C}<math> is given, then the definition can be generalized to
- <math>(\Delta_\gamma\phi)(v)=\sum_{w:\textrm{dist}(w,v)=1}\gamma_{wv}(\phi(w)-\phi(v))<math>
where <math>\gamma_{wv}<math> is the weight value on the edge <math>wv\in E<math>.
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
- <math>\frac{\partial^2F}{\partial x^2} =
\lim_{\epsilon \rightarrow 0}
\frac{(F(x+\epsilon)-F(x))+(F(x-\epsilon)-F(x))}{\epsilon^2}.
<math>
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 <math>P:V\rightarrow\mathbb{R}<math> be a potential function defined on the graph. Note that P can be considered to be a multiplicative operator acting diagonally on <math>\phi<math>
- <math>(P\phi)(v)=P(v)\phi(v).<math>
Then <math>H=\Delta+P<math> 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
- <math>G(v,w;\lambda)=\langle\delta_v| \frac{1}{H-\lambda}| \delta_w\rangle <math>
where <math>\delta_w<math> is understood to be the Kronecker delta function on the graph: <math>\delta_w(v)=\delta_{wv}<math>; that is, it equals 1 if v=w and 0 otherwise.
For fixed <math>w\in V<math> and <math>\lambda<math> a complex number, the Green's function considered to be a function of v is the unique solution to
- <math>(H-\lambda)G(v,w;\lambda)=\delta_w(v).<math>
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