We consider a Schrdinger particle on a graph consisting of <i>N</i> links joined at a single point. Each link supports a real locally integrable potential <i>V</i> _<i>j</i>; the self-adjointness is ensured by the type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as <i>x</i> ¹ along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrdinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the coupling constant may be interpreted in terms of a family of squeezed potentials.

In this paper, we attempt to reconstruct one of the last and incomplete projects of Volodya Geyler. We study the motion of a quantum particle in the plane to which a halfline lead is attached, assuming that the particle has spin and the plane component of the Hamiltonian contains a spin-orbit interaction, of Rashba or Dresselhaus type. We construct a class of admissible Hamiltonians and derive an explicit expression for the Green function, which is applied to scattering in a system of this kind.

We consider an open quantum dot modeled by a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, the system exhibits embedded eigenvalues. They turn into resonances if the symmetry is violated, either by a magnetic field or by deformation of the well. We construct a perturbation theory of these resonances in the case of a weak perturbation and discuss other properties of the model.

We consider scattering of a three-dimensional particle on a finite family of potentials. For some parameter values the scattering wavefunctions exhibit nodal lines in the form of closed loops, which may touch but do not entangle. The corresponding probability current forms vortical singularities around these lines; if the scattered particle is charged, this gives rise to magnetic flux loops in the vicinity of the nodal lines. The conclusions extend to scattering on hard obstacles or smooth potentials.

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.