We consider a two-dimensional particle of charge e interacting with a homogeneous magnetic eld perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to 2 sgne times the number of ux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum e ect since the corresponding Hannay angle is zero.

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.

Effect of the spin-involved interaction of electrons with impurity atoms or defects to the transport properties of a two-dimensional electron gas is described by using a simplifying two-component model. Components representing spin-up and spin-down states are supposed to be coupled at a discrete set of points within a conduction channel. The used limit of the short-range interaction allows to solve the relevant scattering problem exactly. By varying the model parameters different transport regimes of two-terminal devices with ferromagnetic contacts can be described. In a quasi-ballistic regime the resulting difference between conductances for the parallel and antiparallel orientation of the contact magnetization changes its sign as a function of the length of the conduction channel if appropriate model parameters are chosen. The effect is in agreement with recent experimental observations.

A product formula is proved which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P.

We discuss a model of a leaky quantum wire and a family of quantum dots described by Laplacian in L²(²) with an attractive singular perturbation supported by a line and a finite number of points. The discrete spectrum is shown to be nonempty, and furthermore, the resonance problem can be explicitly solved in this setting; by Birman-Schwinger method it is reformulated into a Friedrichs-type model.