A “Hybrid Plane” with spin-orbit interaction

In this paper, we attempt to reconstruct one of the last and incomplete projects of Volodya Geyler. We study the motion of a quantum particle in the plane to which a halfline lead is attached, assuming that the particle has spin ½ and the plane component of the Hamiltonian contains a spin-orbit interaction, of Rashba or Dresselhaus type. We construct a class of admissible Hamiltonians and derive an explicit expression for the Green function, which is applied to scattering in a system of this kind.     

Open quantum systems and Feynman integrals: Some problems

A few open problems of mathematical physics are presented. They concern open quantum systems and Feynman path integrals; some of them are technical, while others are of conceptual importance.Dedicated to the 30th anniversary of the Joint Institute for Nuclear Research.     

Curvature vs. Thickness in quantum waveguides

It is known that there is at least one bound state in a curved quantum waveguide provided it is sufficiently thin. In this paper we investigate the critical thickness of two-dimensional waveguides above which the discrete spectrum becomes void. We have found an expression for it in the case of a small bending angle. In the general case, its asymptotic behaviour with respect to the bending angle is shown to be governed by local smoothness properties of the boundary. Uniqueness of the critical thickness is also discussed.Dedicated to the memory of M. Gmitro.One of the authors (P.E.) is grateful for the hospitality extended to him at the University of Toulon and C.N.R.S. Marseille where the most part of this work was done. A partial support by Czechoslovak Grant Agency under Contract No. 14814 is acknowledged.     

Highest-weight representations ofsl(2, ℂ) andsl(3, ℂ) via canonical realizations

The infinite-dimensional representations of thesl(n+1, ) Lie algebras (maximal representations) constructed in our previous paper are studied on the two simplest examplesn = 1,2. The sufficient condition for irreducibility of the maximal representations is proved to be also necessary in these cases. It is further shown, that our method allows us to construct other set of infinite-dimensional highest-weight representations ofsl(3, ), so calledmixed representations which are irreducible in some cases when the maximal as well as the standard highest-weight representations (Verma modules) are reducible.Dedicated to the 25th anniversary of the Joint Institute for Nuclear Research.The authors are grateful to Prof. A. A. Kirillov, Dr. A. U. Klimyk, Dr. W. Lassner and Prof. D. P. Zhelobenko for stimulating discussions.     

Canonical realizations of classical lie algebras

We construct sets of canonical realizations for all classical Lie algebras (A _n ,B _n ,C _n ,D _n ). These realizations depend ond parameters,d=1, 2, 3,...,n; all Casimir operators are realized by multiples of identity. For most of the real forms of these algebras we give sets of realizations which are, moreover, in well-defined sense skew-Hermitian. Further we study extremal cases of the presented realizations. The realizations with minimal numbers of canonical pairs are discussed from the point of view of general results concerning minimal realizations. On the other hand, a connection is found between our maximal realizations ofA _n and the Gel'fand-Kirillov Conjecture.The authors would like to thank Prof. A.Uhlmann for his kind interest in this work. They are very grateful to Prof. A. A.Kirillov and Prof. D. P.Zhelobenko for helpful discussions and to Prof. J.Dixmier for his informative letter concerning the problem mentioned in Sect. 5.One of the authors (W. L.) thanks Prof. I.Úlehla for the hospitality at the Nuclear Center of the Charles University, Praha.     

A simple model of thin-film point contact in two and three dimensions

We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two-parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admissible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films.On leave of absence fromNuclear Physics Institute, Czechosl. Acad. Sci., 250 68 e near Prague, Czechoslovakia.On leave of absence fromNuclear Centre, Charles University, V Holeovikách 2, 180 00 Prague 8, Czechoslovakia.     

Two-component interference effect: model of a spin-polarized transport

The effect of spin-involved interaction on the transport properties of disordered two-dimensional electron systems with ferromagnetic contacts is described using a two-component model. Components representing spin-up and spin-down states are supposed to be coupled at a discrete set of points. We have found that due to the additional interference arising in two-component systems the difference between conductances for the parallel and antiparallel orientations of the contact magnetization changes its sign as a function of the length of the conducting channel.