I'm working on time-independent degenerate perturbation theory for the Hydrogen first excited state. I have the following perturbation $H$:
$H = \lambda V_0 \sin^2 \theta \sin 2\phi = \lambda V$.
We can see that $P V P^{-1} = V$ so this means that $\langle n l m | V | n' l' m'\rangle$ might be $\neq 0$ if and only if $l+l'$ is an even integer.
We can also see that $\int_0^{2 \pi} \sin 2\phi \ \mathrm{d}\phi = \int_0^{2 \pi} \sin 2\phi \ e^{\pm i \phi} \mathrm{d}\phi = 0$, so we have to add that $\langle n l m | V | n' l' m'\rangle$ might be $\neq 0$ if and only if $m = -m' = \pm 1$.
So I arrive to the conclussion that $\langle n l m|V|n'l'm'\rangle = 0$ except from $\langle 211|V|21-1\rangle$ and $\langle21-1|V|211\rangle$.
But we can see that $\langle 211 |V| 21-1\rangle = -\langle21-1|V|211\rangle = V_0 \frac{(2a)^{-3}}{8 \pi a^2} \int_0^{\infty} r^4 e^{-r/a} \mathrm{d}r \int_0^{\pi} \sin^5 \theta \ \mathrm{d}\theta \int_0^{2\pi} e^{-2i\phi} \sin 2\phi \ \mathrm{d}\phi = \frac{2 V_0}{5} i$ which is a complex number.
What am I doing wrong or what is the interpretation for these results?
When I construct the matrix $\langle nlm |V| n'l'm'\rangle$ and diagonalize it I get 3 eigenvalues $0, \frac{2}{5}, \mathrm{and}\ -\frac{2}{5}$ in terms of $\lambda V_0$, so everything's fine in this way.
Thanks in advance, please help a confined quantum mechanics student!!!