Quantum systems with balanced gain and loss, signatures of branch points, and dissociation effects

Gain and loss to the wave function of quantum mechanics can in a convenient way be modelled by effective non-Hermitian Hamiltonians. Imaginary contributions to the potential introduce source and drain terms for the probability amplitude. A special class of non-Hermitian Hamiltonians are those which possess a parity-time symmetry. In spite of their non-Hermiticity these Hamiltonians allow for real energy eigenvalues, i.e. the existence of stationary states in the presence of balanced gain and loss. This effect has been identified theoretically in a large number of quantum systems. Its existence has also been proved experimentally in coupled optical wave guides. The wave guides are, however, only optical analogues of quantum systems. In the first part of this thesis it is shown from the theoretical side that Bose-Einstein condensates in a double-well setup are an ideal candidate for a first experimental realisation of a genuine quantum system with parity-time symmetry. When particles are removed from one well and coherently injected into the other the external potential is parity-time symmetric. To investigate the system the underlying time-independent and time-dependent Gross-Pitaevskii equations are solved numerically. It turns out that a subtle interplay between the nonlinearity of the Gross-Pitaevskii equation and the gainloss effect leads to a complicated dynamics of the condensate wave function. However, the most important result is the existence of stationary states that are sufficiently stable to be observable in an experiment. Two suggestions for experimental realisations are presented. They are based on the idea of embedding the non-Hermitian parity-timesymmetric system into a larger structure described by a Hermitian Hamiltonian. A further effect of non-Hermitian Hamiltonians are so-called exceptional points, at which two resonances coalesce such that both their eigenvalues and wave functions become identical. It is shown that an exceptional point can unambiguously be identified by a characteristic non-exponential decay of the resonances. With numerically exact calculations for the hydrogen atom in crossed electric and magnetic fields this effect is verified in an experimentally accessible quantum system. The second part of the thesis is devoted to semiclassical Gaussian approximations to the Boltzmann operator, which have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. A numerically cheap frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters is developed. It is applied to the cases of Ar3 and Ar6. For these clusters classical-like transitions in one step from a bounded moiety to free particles are found for increasing temperatures. Additionally, the structure of the Ar6 cluster is studied in the bound configuration and during the dissociation. Quantum