For the past few days, I have been struggling to convey to mathematica to solve a PDE that is in terms of the independent variables $(z,\bar{z})$. I know mathematica supports solving PDEs with respect to $(x,y)$, but that conversion would be very very tedious.
The PDE I am looking at is an eigenvalue equation. I'm looking for the plane wave solutions of $$ (-\Delta^{(H)}+\epsilon \hat{Q})\psi(z,\bar{z})= E \psi(z,\bar{z})$$ where $$\Delta^{(H)}= -(1-|z|^2)^2 \cdot \underbrace{4\frac{\partial}{\partial z}\frac{\partial}{\partial \bar{z}}}_{=\Delta}$$ is called the hyperbolic Laplacian, $\Delta = \nabla^2$ refers to the standard Laplacian in $\mathbb{R}^2$ and $\epsilon>0$ is a small parameter.
Denote by $$\frac{\partial}{\partial z}:= \partial_{z}\ ;\ \frac{\partial}{\partial \bar{z}}:= \bar{\partial_{z}}$$
The Operator $\hat{Q}$ is as follows $$\hat{Q}= \partial_{z}^2 \left((1-|z|^2)^3\partial_{z}\right)+\bar{\partial_{z}}^2 \left((1-|z|^2)^3\bar{\partial_{z}}\right)$$
That is the full problem I'm trying to solve but I wanted to start with a test case first. I am currently struggling to have mathematica consider $z,\bar{z}$ independently yet retain the fact that $z,\bar{z}\in\mathbb{C}$ and then solve the following eigenvalue problem.
Consider the simpler eigenvalue equation for which I know the solutions
$$-\Delta^{(H)} \psi=\varepsilon_{K}\psi$$ With $K=ke^{i\beta}\in\mathbb{C}$, The eigenfunctions have been analytically found to be $$\psi(z)=\left(\frac{1-|z|^2}{|1-ze^{-i\beta}|^2}\right)^{\frac{1+ik}{2}}$$ that satisfy the eigenvalue equation $$-\Delta^{(H)} \psi=\varepsilon_{K}\psi$$ with eigenvalue $\varepsilon_{K}=-(1+k^2)$.
I have verified that this is indeed the eigenfunctions however, It seems I am having trouble telling mathematica to solve this equation as a test case.
I have tried
eqn = -(1 - Abs[z]^2)^2*4*
D[D[u[z, Conjugate[z]], Conjugate[z]], z];
{vals, funs} =
NDEigensystem[{eqn},
u[z, Conjugate[z]], z \[Element] Disk[], 10];
but it is of no avail. I wanted to ask if I could get any help on this matter. Any help would be greatly appreciated!