I have encountered this operator:
$$H := \nabla^2 + \alpha \frac{1}{r^2} $$
with $\alpha \in \mathbb{C}$. Surely, the eigenfunctions are some unique functions defined by the equation $$H \psi = E \psi$$ just like with the Airy equation. I wasn't able to find anything online (mostly because I don't know the name of the equation) so I'd appreciate someone pointing me the right way.
Background on the problem:
The Zeeman interaction of a magnetic monopole reads $\mathcal{H_Z} = \beta r^{-3} \vec{r} \cdot \vec{\sigma}$ where $\vec{\sigma}$ is a vector of Pauli matrices. Encomporating the spin related constants into $\beta$ gives us $\alpha$ and adding kinetic energy results in $\nabla^2$.
Look in Landau and Lifshitz, there is a problem there, known as the motion of a particle in the potential diverging as $1/r^{2}$ near the origin, they solve it there exactly. In fact the solution is the same as just for a free 3D particle in spherical coordinates, as you will get the $l(l+1)/r^{2}$ term from the spherical harmonics, you can define $l(l+1)+\alpha=\beta(l)(\beta(l)+1)$, thus, the most general solution will be like $$\Psi(r, \theta, \varphi)=\int{dk}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}c_{lm}(k)j_{\beta(l)}(kr)Y_{l}^{m}(\theta, \varphi)$$ Where $j_{\beta(l)}(kr)$ is the spherical bessel function of the first kind and $Y_{l}^{m}(\theta, \varphi)$ is the spherical harmonics.