We discuss discuss spectral and scattering properties of a particle confined to a straight Dirichlet tube in R''with a family of point interactions.

We consider an electron constrained to move on a surface with revolution symmetry in the presence of a constant magnetic field B parallel to the surface axis. Depending on B and the surface geometry the transverse part of the spectrum typically exhibits many crossings which change to avoided crossings if a weak symmetry breaking interaction is introduced. We study the effect of such perturbations on the quantum propagation. This problem admits a natural reformulation to which tools from molecular dynamics can be applied. In turn, this leads to the study of a perturbation theory for the time dependent BornOppenheimer approximation.

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Resonance and decay phenomena are ubiquitous in the quantum world. To understand them in their complexity it is useful to study solvable models in a wide sense, that is, systems which can be treated by analytical means. The present review offers a survey of such models startingt he classical Friedrichs result and carryingfu rther to recent developments in the theory of quantum graphs. Our attention concentrates on dynamical mechanism underlyingre sonance effects and at time evolution of the related unstable systems.