Can we solve the Helmholtz Equation for Gabriel's Horn: $$G = \left\{ {\sqrt{x^2 + y^2}} \leq \frac{1}{z} \text{ and } 1 \leq z\ \ ;\ \ \{x,y,z\} \in \mathbb{R}^3\right\}, \quad \nabla^2 u|_G = \lambda u|_G, \ u|_{\partial G} = 0$$
I put the $1 \leq z$ condition in more-or-less arbitrarily, I was worried about bad things happening near $z=0$; but perhaps this isn't really an issue. As for what I've tried - I don't really know any PDE methods apart from `try to guess an answer', I've used mathematica to generate some of the results for a 2D-slice of this Gabriel's horn:
using this answer - and I was more or less just interested in what we can say analytically.