I want to write a python simulation that requires the eigenfunctions and eigenvalues of the Laplace-Beltrami operator over the torus:
$$ x=(R-rcos\theta)cos\phi , y=(R-rcos\theta)sin\phi , z=rsin\theta ,r<R,\theta ,\phi \in [0,2\pi] $$
I know that the Laplace-Beltrami operator is: $$ \nabla^2 u= \frac{1}{r^2(R-rcos\theta)}*[(R-rcos\theta) u]_{\theta\theta} +\frac{1}{(R-rcos\theta)^2}u_{\phi\phi} $$
Can you help me find information about the eigenfunctions and eigenvalues of this operator? I searched in many papers, but I didn't find a complete answer...
In addition - is there any python package that implements those eigenfunctions and an expansion function for this basis?
Thanks!