In this paper we investigate the operator H=??(??) in L2 (R2), where> 0 and? is a closed C4 Jordan curve in R2. We obtain the asymptotic form of each eigenvalue of H as tends to in nity. We also get the asymptotic form of the number of negative eigenvalues of H in the strong coupling asymptotic regime. MSC: 35J10, 35P15

Point interactions described by formal shaped potentials were used in quantum mechanics from the early thirties. It lasted several decades, however, until a proper way to handle the corresponding Schrdinger operators was found. Following the observation of Berezin and Faddeev [4], the problem was studied systematically in the eighties. The results are summarized in the monograph [1], references to some recent work are given, eg, in [2, 8]. In its present form the pointinteraction method represents a versatile tool for constructing solvable models whose power has not been yet, to our opinion, appreciated fully in the physical community. Within the standard quantum mechanical formalism point interactions can be constructed as long as the configuration space dimension does not exceed three, and they can be interpreted as limits of suitable families of squeezed potentials. In the onedimensional case the limiting argument admits a straightforward interpretation, because the pointinteraction coupling constant is just the integral of the approximating potential: a slow particle on the line with a wellspread wavefunction sees only the average value of a localized potential. In distinction to that the approximation by scaled potentials in dimension two and three requires existence of a zeroenergy resonance and a particular couplingconstant renormalization; a detailed discussion of the related mathematical problems can be found in [1, 9]. This gives these point interactions a flavour of something exceptional, for instance, one is led to believe that they cannot be used to model welllocalized repulsive potentials. That would have consequences

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.