Multipole matrix elements of Green function of Laplace equation

TL;DR

This paper presents a method to calculate multipole matrix elements of the Green function for the Laplace equation, simplifying the process using Fourier and Hankel transforms based on symmetry.

Contribution

It introduces a novel approach to compute multipole matrix elements leveraging Fourier space simplifications and rotational symmetry, enhancing computational efficiency.

Findings

Derived explicit formulas for multipole matrix elements.

Simplified three-dimensional Fourier transforms to one-dimensional Hankel transforms.

Applicable to electrostatics problems involving spherical charge distributions.

Abstract

Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation.