Arar, Nouria; Boulmezaoud, Tahar Z. Eigenfunctions of a weighted Laplace operator in the whole space. (English) Zbl 1308.35151 J. Math. Anal. Appl. 400, No. 1, 161-173 (2013). Summary: We study the spectrum of the weighted Laplacian \(\varrho^{ - 1}\Delta \) in the whole space \(\mathbb R^{n}\). We prove, under adequate conditions on \(\varrho^{ - 1}\), that this spectrum is discrete and we derive an explicit formula for eigenvalues and eigenfunctions when \(\varrho^{ - 1}=(|x|^{2}+1)^{2}\). We get by the way a complete family of rational functions which are mutually orthogonal in a weighted \(L^{2}\) space. Cited in 5 Documents MSC: 35P05 General topics in linear spectral theory for PDEs 47A10 Spectrum, resolvent 35J15 Second-order elliptic equations Keywords:Laplace operator; unbounded domains; rational functions; stereographic projection; weighted spaces PDFBibTeX XMLCite \textit{N. Arar} and \textit{T. Z. Boulmezaoud}, J. Math. Anal. Appl. 400, No. 1, 161--173 (2013; Zbl 1308.35151) Full Text: DOI