Is it possible to find an analytical solution for eigenvalues and eigenfunctions of the Dirichlet Laplacian on a solid torus? More precisely, is it possible to solve analytically the following problem:
$$ \begin{cases} \Delta u = \lambda u \\ u|_{\partial T} = 0 \end{cases} $$
where T is a solid torus ($T=S^1 \times D$, $S^1$ is a circle and D is a disk in $\mathbb{R}^2$). I have found a method of separation of Laplace’s equation in toroidal coordinates in "Alternative separation of Laplace’s equation in toroidal coordinates and its application to electrostatics" (https://doi.org/10.1016/j.elstat.2005.11.005), but I'm not quite sure if it is applicable in this case (the eigenvalue problems are not considered in this article).