Spectral inequalities on manifolds of negative curvature

This thesis is devoted to the study of the Laplace-Beltrami operator in hyperbolic space and the Schrödinger operators it produces. The text can be divided into four parts. In the first part comprising chapters 2 to 5 certain notions and classic theorems from the spectral theory of self-adjoint operators in general and of Schrödinger operators in particular are reviewed. Particular attention is paid to Lieb-Thirring inequalities in euclidean space. The second part, which comprises chapter 6, constitutes the core of this thesis. Here it is shown that many Lieb-Thirring inequalities that hold in euclidean space have analogues in hyperbolic space. The third part, chapter 7, uses elementary methods to prove a Weyl law for the hyperbolic Laplacian on domains. Though such a result can be obtained using different established approaches, our proof shows the utility of the method of coherent states. Finally, in the fourth part, chapter 8, a Weyl law is obtained for Schrödinger operators having a logarithmic differential part and an increasing potential that is bounded from below.