the Helmholtz equation $$\Delta \psi + k^2 \psi = 0$$ has a lot of fundamental applications in physics since it is a form of the wave equation $\Delta\phi - c^{-2}\partial_{tt}\phi = 0$ with an assumed harmonic time dependence $e^{\pm\mathrm{i}\omega t}$.
$k$ can be seen as some kind of potential - the equation is analogue to the stationary Schrödinger equation.
The existance of solutions is to my knowledge linked to the separability of the Laplacian $\Delta$ in certain coordinate systems. Examples are cartesian, elliptical and cylindrical ones.
For now I am interested in a toroidal geometry, $$k(\mathbf{r}) = \begin{cases} k_{to} & \mathbf{r}\in T^2 \\ k_{out} & \text{else}\end{cases}$$
where $T^2 = \left\{ (x,y,z):\, r^2 \geq \left( \sqrt{x^2 + y^2} - R\right)^2 + z^2 \right\}$
Hence the question:
Are there known solutions (in terms of eigenfunctions) of the Helmholtz equation for the given geometry?
Thank you in advance
Sincerely
Robert
Edit: As Hans pointed out, there might not be any solution according to a corresponding Wikipedia article. Unfortunately, there is no reference given - does anyone know where I could find the proof?