We consider a modification of the Winter model describing a quantum particle in presence of a spherical barrier given by a fixed generalized point interaction. It is shown that the three classes of such interactions correspond to three different types of asymptotic behaviour of resonances of the model at high energies.

The subject of the paper is Schrdinger operators on tree graphs which are radial, having the branching number bn at all the vertices at the distance tn from the root. We consider a family of coupling conditions at the vertices characterized by (bn1)2+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of {tn+1tn} growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrdinger operator can be purely absolutely continuous.

We discuss differences between the exact<i>S</i>-matrix for scattering on serial structures and a known factorized expression constructed of single-element<i>S</i>-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities.

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.

We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a hybrid surface consisting on a halfline attached by its endpoint to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term.