May be who knows about the tranformations from one diagonal Laplacian to another diagonal Laplacian (change of variables)? Especially I'm interested in a case of transformation of Cartesian Laplacian to diagonal Laplacian with rational coefficients in 3D case.
Volodya, There are a lot of transformations which you need are well known. For instance, transformations in 2D from Cartesian coordinates to r-phi in the cylindrical system.
Evgenii Alexandrovich, I know few examples of such pairs of the diagonal matrix. But I need in a general theory. See attached file.
You could read on tensors and its application to curvilinear coordinates for clues I think.
In your file, it wants you to define the metric of tensor transformations but I am not acquainted with stackel coordinates to be of any help.
I am not sure this will help, but for a Riemannian manifold, you have the related Laplace Beltrami operators here (when the metric changes conformally). Of course, there are several papers in the topic. This responds to the first part of your question, and not to your particular case
https://anhngq.wordpress.com/2014/12/31/conformal-change-of-the-laplace-beltrami-operator/
Best wishes :-) Hope it helps.
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