Green Function of the Poisson Equation: D=2,3,4

Abstract

We study the Green function of the Poisson equation in two, three and four dimensions. The solution g of the equation nabla^2 g(x - x') = delta^(D)(x - x'), where x and x are D-dimensional position vectors, is customarily expanded into radial and angular coordinates. For the two-dimensional case (D=2), we find a subtle interplay of the necessarily introduced scale L with the radial component of zero magnetic quantum number. For D=3, the well-known expressions are briefly recalled; this is done in order to highlight the analogy with the four-dimensional case, where we uncover analogies of the four-dimensional spherical harmonics with the familiar three-dimensional case. Remarks on the SO(4) symmetry of the hydrogen atom complete the investigations.