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

To describe quantum oscillations on metallic rings and similar effects, some authors introduced recently an ideal device which splits the electron wavefunction at the junction of three wires. A proper quantum-mechanical treatment requires, however, that the splitting procedure is described by a hamiltonian. In this Letter we show how this can be achieved in the framework of the self-adjoint-extension theory.

We consider resonant vortices around nodal points of the wave function describing electron transport through a mesoscopic device. With a suitable choice of the device geometry, the dominating role is played by single vortices with a preferred orientation. To characterize the strength of the resulting magnetic moment, we have introduced a magnetance, the quantity defined in analogy with the device conductance. Its basic properties and possible experimental detection are discussed.

This is the first of a series of papers in which we treat in detail a simple non-relativistic model of a spinless particle decaying into two lighter particles, which is similar to the Lee-model description of V-particle decay. Galilean covariance is formulated properly by means of a unitary projective representation acting on the state space of the model.