In the spectral theory it does make a difference, whether we consider differential operators on bounded or unbounded domains. In order to treat eigenvalue problems on the whole Euclidean space, we construct Sobolev spaces over mathbb {R}^{n}, which are weighted by the Gaussian normal distribution. By the methods presented in Chapters 2, 8, and 10 of the treatise F. Sauvigny: Partial Differential Equations 1 and 2, Springer Universitext (2012), we can prove an analogue of the Sobolev embedding theorem and a Rellich selection theorem for the Sobolev spaces W_{0}^{1,p}(mathbb {R}^{n},gamma ) weighted by γ- with vanishing values towards infinity. We achieve these specific results for our entire Sobolev spacesW^{1,p}(mathbb {R}^{n},gamma ), since we concentrate on the Gaussian normal distribution γ as our weight function. Even our notion of the weighted partial derivative depends on this weight function. Within the so-called Gauß–Rellich spaceW_{0}^{1,2}(mathbb {R}^{n},gamma ) we shall investigate the discrete spectrum of weighted elliptic operators over mathbb {R}^{n} by spectral methods. There we rely on the treatise F. Sauvigny: Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren, Springer Spektrum (2019). By reflection methods, we solve eigenvalue problems for elliptic differential operators on the sectorial domain mathbb {R}_{+}^{n} under vanishing and mixed boundary conditions.