Editing Discrete Laplace operator (section)

==Discrete Schrödinger operator==
Let <math>P\colon V\rightarrow 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 equation|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 [[spectrum of an operator|spectral properties]] of this Hamiltonian can be studied with [[Stone space|Stone's theorem]]; this is a consequence of the duality between [[poset]]s and [[Boolean algebra (structure)|Boolean algebra]]s.

On regular lattices, the operator typically has both traveling-wave as well as [[Anderson localization]] solutions, depending on whether the potential is periodic or random.

The [[Green's function]] of the discrete [[Schrödinger operator]] is given in the [[resolvent formalism]] by 
:<math>G(v,w;\lambda)=\left\langle\delta_v\left| \frac{1}{H-\lambda}\right| \delta_w\right\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>