# [Physics] Finding Green function using eigenfunction expansion method

boundary conditionselectrostaticsgreens-functionshomework-and-exercises

Given the Dirichlet boundary condition, I am to show that the functions that satisfy
$$(\nabla ^2 + k_{lmn}^2) \psi_{lmn} (x,y,z) = 0$$
are given by
$$\psi_{lmn} = (\frac{\pi}{2x})^{1/2} J_{l+1/2}(x) Y_{lm} (\theta, \phi)$$
for a hollow sphere or radius $a$, where the green function $G$ can be expanded as $$G(\vec{x},\vec{x'}) = \sum_n a_n(\vec{x'}) \psi_n (\vec{x}).$$

Note that I can use the solution of the Helmhotz equation, which is given to me as

$$\Psi = \left\{ \begin{array}{c} J_m(\rho \sqrt{k^2-\alpha^2}) \\ Y_m(\rho \sqrt{k^2 – \alpha^2}) \end{array} \right\} \left\{ \begin{array}{c} e^{i\alpha z} \\ e^{-i \alpha z} \end{array} \right\} \left\{ \begin{array}{c} e^{i m \phi} \\ e^{-i m \phi} \end{array} \right\}$$

where the brackets express a linear combination of their arguments.

I am confused as to how to proceed from there and how to use the BC's to make it look like the desired result.

I argee with answers above: you don't need Green's function to solve this equation, you just need to separate variables by entering spherical coordinate system like this $\psi(x,\theta,\phi)=R(r)\psi_{ang}(\theta,\phi)$. After doing this you'll get two separate equations on functions $R(r)$ and $\psi_{ang}(\theta,\phi)$: Bessel equation for $R(r)$, see Bessel functions and equation for angles. Solution of second equation is represented by spherical harmonics.
P.S. It is also easier to search for $R(r) = \frac{\chi(x)}{x}$. Than you'll exactly get Bessel equation.