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Eigenfunctions of a weighted Laplace operator in the whole space. (English) Zbl 1308.35151

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.

MSC:

35P05 General topics in linear spectral theory for PDEs
47A10 Spectrum, resolvent
35J15 Second-order elliptic equations
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