Discrete delta-potential method

The discrete delta potential method is a combination of both numerical and analytic method used to solve the Schrödinger equation the main feature of this method is to obtain first a discrete approximation of the potential in the form:

V(r) =	∑	V(ai))δ(r − ai)
	i

where the i index runs over several discrete chosen points of the continuous potential, with this we can solve the SE in the form:

Φ(s) =	∑	G(s,ai)V(ai)Φ(ai) + γ(s)
	i

Where G(s, r) is the Green function associated to the free particle Hamiltonian H0, setting s=ai for every i would give a system of linear equation.

For a better approach if potential V(x) is weak we can treat the continuous part of the potential and perform perturbation theory to obtain a better wave function.

If the points are equidistant : | a_{i + 1} − a_i | = h the wave function will depend on h so performing the solution to several h and extrapolating to the value h=0 we could obtain the solution of SE (Schrödinger equation).

This method is valuable when we cannot perform perturbation theory and need some analytic approach to the solution of SE:HΦ = E_nΦ