Today, we consider those PDEs with Laplacian in spherical coordinate.

$$ \Delta u = \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d u}{\d r} \right) + \frac{1}{r^2} \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d u}{\d \theta}) + \frac{1}{r^2 \sin^2 \theta} \frac{\d^2 u}{\d \phi^2}. $$

Notice that $\theta$ and $\phi$ varies in a bounded domain, hence the eigenvalue problems for these variables have discrete eigenvalues.

We look for eigenfunctions of the form $$ u (r, \theta, \phi) = R® \Theta(\theta) \Phi(\phi). $$ Then $ \Delta u = \lambda u$ is equivalent to $$ \frac{1}{u} \Delta u = \frac{1}{R} \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + \frac{1}{\Theta} \frac{1}{r^2} \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d \Theta }{\d \theta}) + \frac{1}{\Phi } \frac{1}{r^2 \sin^2 \theta} \frac{\d^2 \Phi}{\d \phi^2}. $$ which in turns breaks into equations $$ \frac{\d^2 \Phi}{\d \phi^2} = \lambda_\phi \Phi $$ $$ \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d \Theta}{\d \theta}) + \frac{\lambda_\phi}{\sin^2 \theta} \Theta = \lambda_\theta \Theta $$ $$ \frac{1}{R} \frac{1}{r^2} \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + \frac{\lambda_\theta}{r^2} = \lambda_r \quad \Rightarrow \left( \frac{\d}{\d r} r^2 \frac{\d R}{\d r} \right) + (\lambda_\theta - \lambda_r r^2) R = 0 $$

The equation about $\Phi$ has $\sin(m\phi), \cos(m\phi)$ as eigenfunctions, with eigenvalues $\lambda_\phi = -m^2$.

The equation about $\Theta$ has associated Legendre polynomial as solution, recall that $y(\theta) = P_l^m (\cos \theta)$ solves equation (Problem 12.10.2) $$ \frac{1}{\sin \theta} \frac{\d }{\d \theta} ( \sin \theta \frac{\d y}{\d \theta}) + \left( l(l+1) - \frac{m^2}{\sin^2 \theta} \right) y = 0 $$ By comparison, we get eigenvalues $\lambda_\theta = l (l+1)$.

The equation about $R$ is related to the spherical Bessel function's equation, $y = j_n®, y_n®$ solves $$ \left( \frac{\d}{\d r} r^2 \frac{\d y}{\d r} \right) + (r^2 - n(n+1) ) y = 0 $$