Abstract: We consider a quantum particle moving in the plane under an influence of potential which can be modelled by delta interaction supported by two parallel straight lines. We discuss spectral properties of such a system with additional assumption that the interaction asymptotically goes to a constant value. For example, we formulate conditions guaranteeing either the existence of discrete eigenvalues or Hardy-type inequalities. Furthermore, we show that for a certain class of systems with a mirror symmetry the embedded eigenvalues phenomena holds. Finally, we show that the embedded eigenvalues turn into resonances after introducing a small perturbation of the mirror symmetry.
The talk is based on a common work with David Krejcirik.