A two-dimensional model of electron motion in a curved quantum wire of length 2<i>D</i> attached to a pair of macroscopic electrodes is studied. The wire is regarded as infinitely thin and supports a potential which is a combination of a constant transversal-mode energy and an attractive curvature-induced term. The system is shown to have at least one resonance which exhibits spectral concentration as <i>D</i>.

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles that the waveguide can bind is given by a one-particle BirmanSchwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.

We discuss approximations of the vertex coupling on a star-shaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the CheonShigehara technique using interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges, one can approximate the {n+1\choose 2}-parameter family of all time-reversal invariant couplings.

We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.

The most general admissible boundary conditions are derived for an idealized AharonovBohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; the other two parameters of the model are the AharonovBohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the AharonovBohm effect with a point interaction. Spectral properties of the derived Hamtonians are studied in detail.