# Green's function expansion in a set of eigenfunction

1. ### kristobal hunta

7
Hi! I encountered the problem that I need to decompose the Green function into a set of eigenfunction. Particularly, I have the free space Green function
$$G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|}$$
and I need to express it into series of cylindrical mode eigenfunctions
$$\Psi ( \vec r; k) = H_m ( q r) sin( h z) e^{i m \phi}$$
$$k^2 = q^2 + h^2, h = \frac { \pi } {2 L}$$

here H - Hankel's function of the first kind.
Eigenfunction forms a complete set, with discrete spectrum of eigenvalues q and h.
I know that we can decompose the Green function into set of eigenfunctions, but I have the Green function for spherical representation, and eigenfunctions are from waveguide formed by two infinite plates parallel to each other. I couldn't find anything relevant about expanding the Green function into arbitrary set of eigenfunctions. Would appreciate any opinion or advice on the matter :)

2. ### Dr Transport

1,507
Look in Jackson's Electrodynamics book, I believe that the solution can be found by applying either chapter 2 or 3's methods.

3. ### waseem

1
i want notes about quantum dynamics(schrodinger,heisenberg and interaction representation or pictures of quantum mechanics)