The work presented in this thesis revolves around spectral inequalities and their applications in quantum mechanics. In Paper A, the ground state energy of an atom confined to two dimensions is analyzed in the limit when the charge of the nucleus Z becomes very large. The main result is a two-term asymptotic expansion of the ground state energy in terms of Z. Paper B deals with Hardy inequalities for the kinetic energy of a particle in the presence of an external magnetic field. If the magnetic field has a non-trivial radial component, we show that Hardy’s classical lower bound can be improved by an extra term depending on the magnetic field. In Paper C we study interacting Bose gases and prove Lieb-Thirring type estimates for several types of interaction potentials, such as the hard-sphere interaction in three dimensions, the hard-disk interaction in two dimensions as well as homogeneous potentials. / <p>QC 20140520</p>