Singular, particle potentials and the Greens function

To numerically resolve the singular, particle potentials we use the wavelet-Galerkin method to find the Greens function for the Laplace equation with point sources at the particle locations. The point sources are defined by the wavelet-Galerkin delta function. The singularities of the Greens function for the Laplace equation correctly correspond to the singularities of the coulomb potential. Therefore, we use the wavelet-Galerkin method in a consistent to both approximate the potential and solve the Schroedinger equation.

The numerical implementation is straight forward. In effect, we expand the solution in periodic, wavelet-Galerkin basis

$$G=\sum\sum G_{i,j}\varphi(x-i)\varphi(y-j)$$

where $\varphi$ is a scaling function. To calculate the Green's Function we resolve the delta function in the space of translates of scaling function

$$\lambda_{x_0,y_0}(x,y)=\sum\sum\varphi(x_0-i)\varphi(x-i) \varphi(y_0-j)\varphi(y-j).$$

Since the translates of the scaling function are orthogonal and complete in $L^2$, the above expression implies that for a square integrable function $f$

$$f(x_0,y_0)=\int\int dxdy \lambda_{x_0,y_0}(x,y) f(x,y),$$

which is the definition of the delta function.

Therefore, we solve, by the wavelet-Galerkin method [47,48], the equation

$$\left(-\Delta + \alpha\right)G(x,\hat x_s;y,\hat y_s)= \sum_s\theta_s\lambda_{x_s,y_s}(x,y)$$

for the partial Green's Function, $G$ with sources at the locations $(x_s, y_s)$ and strengths $\theta_s$. The evaluation of $G$ requires only one solution of the periodic, fast, wavelet-Galerkin solver [46].