I have Schrodinger equation of the form: $$ \left[\frac{\partial^{2}}{\partial{r}^{2}}+\frac{1}{r} \frac{\partial^{}}{\partial{r}^{}}+\frac{1}{r^{2}}\left(\frac{\partial^{}}{\partial{\varphi}^{}}+i\alpha r^{2}\right)^2+ \frac{\partial^{2}}{\partial{z}^{2}}\right] \Psi = -\lambda^{2}\Psi $$
where the Boundary conditions are:
$\Psi(r,\varphi,z) = 0 \ni \ r\geq r_{0}$ and $z \in (-\infty,-z_{0}] \ \cup \ [z_{0},\infty) \ $ i.e. $\Psi = 0 $ on the surface of a cylinder of radius $r_{0}$ and height $2z_{0}$.
$\Psi(r,\varphi+2\pi,z)=\Psi(r,\varphi,z) \ \forall \ \varphi \in [0,2\pi)$
Is there a general method to solve for the eigenvalues and eigenfunctions of such an PDE with Dirichlet BCs ?
I have tried to reduce the problem into a single variable by considering $\Psi$ to be separable and then performing a Fourier transform for the functions in $\varphi,z$.
This gives me the following differential equation: $$ \left[\frac{\partial^{2}}{\partial{r}^{2}}+\frac{1}{r} \frac{\partial^{}}{\partial{r}^{}}+\frac{1}{r^{2}}\left({k_{\varphi}}+i\alpha r^{2}\right)^2+ k_{z}^{2}\right] R(r) = -\lambda^{2} R(r) $$ In this case the problem takes a form much similar to the Bessel equation. Can it be further modified to be transformed into the general Bessel equation to find the eigenvalues and eigen functions.