Eigenfunctions of a weighted Laplace operator in the whole space

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