Elliot Backman: The spectrum of the Laplace-Beltrami operator on noncompact manifolds

In this thesis we introduce the Laplace-Beltrami operator on Riemannian manifolds and study its spectrum on noncompact manifolds. For compact connected manifolds it is known that the Laplace-Beltrami operator has a discrete spectrum consisting of real spectral points, but in the case of noncompact manifolds the picture becomes more complicated. The manifolds \(\mathbb{R}^n\) and \(\mathbb{H}^n\), whose spectra are shown to be \([0, \infty)\) and \([(n-1)^2/4, \infty)\) respectively, serve as examples of this. We also find that the spectrum of the wave operator, that is the operator corresponding to the Laplace-Beltrami operator on Minkowski space, is \(\mathbb{R}\). In order to calculate these spectra we present the notion of approximate eigenvalue sequences and clarify how these sequences relate to the spectrum of a linear operator.