Table of contents

This paper looks at a magnetic Shrödinger operator on a graph of special form in . It is called an armchair graph because graphs of this form with operators on them are used as a possible model for the so-called armchair nanotube in the homogeneous magnetic field which has amplitude  and is parallel to the axis of the nanotube. The spectrum of the operator in question consists of an absolutely continuous part (spectral bands, separated by gaps) and finitely many eigenvalues of infinite multiplicity. The asymptotic behaviour of gaps for fixed  and high energies is described; it is proved that for all values of , apart from a discrete set containing , there exists an infinite system of nondegenerate gaps  with length as . The dependence of the spectrum on the magnetic field is investigated and the existence of gaps independent of  is proved for certain special potentials. The asymptotic behaviour of gaps as is described.

Bibliography: 32 titles.

A parameter-dependent completely continuous map is considered. The acyclicity of the set of fixed points of this map is proved for some fixed value of the parameter under the assumption that for close values of the parameter the map has a unique fixed point. The results obtained are used to prove the acyclicity of the set of fixed points of a 'nonscattering' map, as well as to study the topological structure of the set of fixed points of an abstract Volterra map.

Bibliography: 13 titles.

Model representations are constructed for a system of bounded linear selfadjoint operators in a Hilbert space  such that

where is a linear operator from  into a Hilbert space  and are some selfadjoint operators in . A realization of these models in function spaces on a Riemann surface is found and a full set of invariants for  is described.

Bibliography: 11 titles.

The asymptotic behaviour of solutions of the first boundary-value problem for a second-order elliptic equation in a domain with angular points is investigated for the case when a small parameter is involved in the equation only as a factor multiplying one of the highest order derivatives and the limit equation is an ordinary differential equation. Although the order of the limit equation coincides with that of the original equation, the problem in question is singularly perturbed. The asymptotic behaviour of the solution of this problem is studied by the method of matched asymptotic expansions.

Bibliography: 11 titles.

Complex Hamiltonian systems with one degree of freedom on  with the standard symplectic structure and a polynomial Hamiltonian function , , are studied. Two Hamiltonian systems , , are said to be Hamiltonian equivalent if there exists a complex symplectomorphism taking the vector field to . Hamiltonian equivalence classes of systems are described in the case , a completed system is defined for , and it is proved that it is Liouville integrable as a real Hamiltonian system. By restricting the real action-angle coordinates defined for the completed system in a neighbourhood of any nonsingular leaf, real canonical coordinates are obtained for the original system.

Bibliography: 9 titles.

The paper examines two examples of multiple orthogonal polynomials generalizing orthogonal polynomials of a discrete variable, meaning thereby the Meixner polynomials. One example is bound up with a discrete Nikishin system, and the other leads to essentially new effects. The limit distribution of the zeros of polynomials is obtained in terms of logarithmic equilibrium potentials and in terms of algebraic curves.

Bibliography: 9 titles.