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Discrete Laplace operator
(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 γwv is the weight value on the edge .
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.
Discrete Green's function
The Green's function of the discrete Schrödinger operator is given in the resolvent formalism by
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 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).
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