The review paper by Grebenkov and Nguyen 2013 analyzes the eigenvalue problem, $$ -\nabla^2 u_n(x) = \lambda_n u_n(x) $$ on a domain $x \in \Omega$. We can also assume Dirchlet boundary conditions, $u_n = 0$ on $\partial \Omega$. The paper suggests that the eigenfunctions and eigenvalues depend only on the domain $\Omega$ and the boundary conditions, rather than the coordinate system used to express the Laplacian. For example, if we consider the spherical shell $\Omega = \left( x \in \mathbb{R}^3 : a \leq ||x|| \leq b \right)$, then the solutions are well known, namely,
$$ u_{nkl} = \begin{pmatrix} j_n(\alpha_{nk} r / b), \\ y_n(\alpha_{nk} r / b) \end{pmatrix} Y_l^m(\theta,\phi) $$
where $j_n,y_n$ are spherical Bessel functions, and the coefficients $\alpha_{nk}$ are determined from the boundary conditions. This solution is obtained by expressing the Laplacian operator in spherical coordinates, $(r,\theta,\phi)$ and doing separation of variables.
Suppose I want to use a different coordinate system, $(r,\alpha,\beta)$ where the angles $(\alpha,\beta)$ also chart the sphere but are nonlinear functions of $(\theta,\phi)$: \begin{eqnarray} \alpha &= \textrm{acos}(\cos{\theta} \sin{\phi}), \\ \beta &= \phi \end{eqnarray}
How would I define an appropriate domain $\Omega$ for this coordinate system? In the case of spherical coordinates, the domain $\Omega$ is defined as surfaces of constant $r$, leading to a spherical shell domain. Would this also be the natural domain to use for my new coordinate system, since the new coordinates also have a radial parameter? Or would the domain be somehow a distorted shell, due to the nonlinear transformation of the angular variables?
In other words, how does one define a natural domain $\Omega$ for a given coordinate system?
Grebenkov and Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Review, 55, 2013.