We investigate an infinite array of point interactions of the same strength in \mathbb{R} ^<i>d</i>, <i>d</i> = 2, 3, situated at vertices of a polygonal curve with a fixed edge length. We demonstrate that if the curve is not a line, but it is asymptotically straight in a suitable sense, the corresponding Hamiltonian has bound states. An example is given in which the number of these bound states can exceed any positive integer.

We study a charged two-dimensional particle confined to a straight parabolic-potential channel and exposed to a homogeneous magnetic field under the influence of a potential perturbation W. If W is bounded and periodic along the channel, a perturbative argument yields the absolute continuity of the bottom of the spectrum. We show that it can have any finite number of open gaps provided that the confining potential is sufficiently strong. However, if W depends on the periodic variable only, we prove by the Thomas argument that the whole spectrum is absolutely continuous, irrespective of the size of the perturbation. On the other hand, if W is small and satisfies a weak localization condition in the the longitudinal direction, we prove by the Mourre method that a part of the absolutely continuous spectrum persists.

We study spectral properties of a spinless quantum particle confined to an infinite planar layer with hard walls, which interacts with a periodic lattice of point perturbations and a homogeneous magnetic field perpendicular to the layer. It is supposed that the lattice cell contains a finite number of impurities and the flux through the cell is rational. Using the Landau-Zak transformation, we convert the problem into investigation of the corresponding fiber operators which is performed by means of Krein's formula. This yields an explicit description of the spectral bands, which may be absolutely continuous or degenerate, depending on the parameters of the model.

We discuss a ring-shaped soft quantum wire modelled by interaction supported by the ring with a generally nonconstant coupling strength. We derive the condition which determines the discrete spectrum of such systems, and analyse the dependence of the eigenvalues and eigenfunctions on the coupling and ring geometry. In particular, we illustrate that a random component in the coupling leads to a localization. The discrete spectrum is also investigated in the situation when the ring is placed into a homogeneous magnetic field or threaded by an AharonovBohm flux and the system exhibits persistent currents.

We discuss a pair of isoperimetric problems which at a glance seem to be unrelated. The first one is classical: one places N identical point charges at a closed curve T at the same arc-length distances and asks about the energy minimum, ie which shape does the loop take if left by itself. The second problem comes from quantum mechanics: we take a Schrdinger operator in