= Solution
For this <charged scalar field>, $\sqrt{-g}=r^3\sin\theta\cos\theta$. On $e^{-i\omega t}$, the <gauge-covariant derivative of a charged scalar field> supplies the shifted frequency
$$
\omega+qA_t=\omega-\frac{qQ}{r^2}.
$$
The wave equation becomes
$$
\frac1{r^3}\frac{d}{dr}\left(r^3f\frac{dR}{dr}\right)+\frac{(\omega-qQ/r^2)^2}{f}R
+\frac{R}{r^2\Theta\sin\theta\cos\theta}\frac{d}{d\theta}\left(\sin\theta\cos\theta\frac{d\Theta}{d\theta}\right)=0.
$$
<Separation of variables> with constant $\lambda$ gives
$$
\boxed{\frac1{\sin\theta\cos\theta}\frac{d}{d\theta}\left(\sin\theta\cos\theta\frac{d\Theta}{d\theta}\right)+\lambda\Theta=0}
$$
and
$$
\boxed{\frac1{r^3}\frac{d}{dr}\left(r^3f\frac{dR}{dr}\right)+\left[\frac{(\omega-qQ/r^2)^2}{f}-\frac{\lambda}{r^2}\right]R=0}.
$$
The angular equation is the axisymmetric <spherical harmonic> equation on the <three-sphere>; regular solutions have $\lambda=\ell(\ell+2)$ with $\ell=0,2,4,\ldots$, the even values selected by the absence of $\phi$ and $\psi$ dependence.
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