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} = 
   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!


1 Answer 1


I will only discuss the test case. An external reference is this PDF by Michael Stone. Some formulas for the Poincare disk model are on this Wikipedia page.

Translation to $x,y$ coordinates. Here $z = x+iy$. Then

DeltaH[f_] := -(1-x^2-y^2)^2*Laplacian[f,{x,y}];

One can see that it is invariant under rotation about the origin, but there is actually a bigger symmetry group, coming from the symmetries of the Poincare disk model, see the PDF above.

Eigenfunctions that OP has given. In $x,y$ coordinates these are

psi[k_,beta_] := ((1-x^2-y^2)/(1+x^2+y^2-2*x*Cos[beta]-2*y*Sin[beta]))^((1+I*k)/2);
lambda[k_,beta_] := -(1+k^2);

Let us check that these are eigenfunctions

- DeltaH[psi[k,beta]] - lambda[k,beta]*psi[k,beta] // Simplify
(* gives 0 *)

The operator is formally selfadjoint on some Hilbert space using an $L^2$-type inner product with measure $dx\,dy / (1-x^2-y^2)^2$. The functions psi[k,beta] are not in that Hilbert space, they have infinite norm. Therefore "eigenfunctions" is to be understood in a loose sense here.

Numerical computation of the eigenvalues. I will use this code:


Here h determines a mesh size, the smaller the better. For DeltaH we must set a=2, but we will also consider the standard Laplacian a=0 and the intermediate value a=1 for comparison.

Standard Laplacian a=0:


The result is quite stable under reduction of mesh size. Compare with known exact values:

(* {5.78319,14.682,26.3746,30.4713} *)

Intermediate case a=1:


We can again compare with exact results: The eigenfunctions include $ (1-r^2) r^{|k|} e^{ik \varphi}$ with integer $k$, whose eigenvalue is $4(1+|k|)$. We have used polar coordinates $x+iy = re^{i\varphi}$.

The case of interest a=2:


We get some numbers, but they are not stable under changing the mesh size. This is expected: This operator has continuous spectrum. One would not expect NDEigensystem to give a meaningful result here. Perhaps someone familiar with NDEigensystem can explain how these numbers are computed.

Formulas. Since OP has mentioned this, here are formulas for the derivatives with respect to $z$ in terms of the partial derivatives with respect to $x$ and $y$:

  • $\begingroup$ Thank so much for your input! This is insightful, however does not answer the main question of solving the PDE with $(z,\bar{z})$ as variables since this is just switching into $(x,y)$ and solving it. $\endgroup$
    – deedeefive
    Aug 5, 2022 at 18:32

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