Eigenfunctions of a weighted Laplace operator in the whole space

We study the spectrum of the weighted Laplacian ϱ−1Δϱ−1Δ in the whole space RnRn. We prove, under adequate conditions on ϱ−1ϱ−1, that this spectrum is discrete and we derive an explicit formula for eigenvalues and eigenfunctions when ϱ−1=(|x|2+1)2ϱ−1=(|x|2+1)2. We get by the way a complete family of rational functions which are mutually orthogonal in a weighted L2L2 space.