Local products for relativistic quantum fields

A definition of local functions of generalized free fields is suggested. Their concrete form is determined in a fairly general setting by analysing realizations of locality. A huge class of new models of relativistic quantum fields results in this way.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Density of the set of attractive compacta

A compact set K in a smooth closed manifold M is said to be attractive, if on M there exists a system of differential equations, for which K is an asymptotically stable invariant set. It is proved that the set of attractive compacta is dense and its complement contains a dense set of type G in the space of all compacta of the manifold M endowed with two natural topologies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 170–175, 1985.     

Errors of solutions of linear algebraic systems

Error estimates are derived for the following methods: the sweepout method for tridiagonal systems, the method of square roots, the bordering method, and the method of reflection matrices. The book of S. K. Godunov is devoted to the last method; he altered the method so that it takes any matrix into a bidiagonal matrix; a considerable part of that book is devoted to the error of this alteration. In the present paper the method of reflection matrices is studied in the form in which it is expounded in the familiar book of D. K. Faddeev and V. N. Faddeeva. Recurrent formulas are obtained for the sweepout method which make it possible to successively estimate errors of the components of the solution vector. In the method of square roots the error of the solution vector is estimated by the quantity Here _i and are small quantities; the first characterizes the accuracy of small arithmetics effects, and the second the round-off error in the reverse step. Further, A is the matrix of the system, m, is its order, f is the vector of free terms, and C and are constants with 0 <1. We shall not present here the rather involved estimates for the bordering method. The error of the solution vector obtained by the method of reflection matrices is estimated by the quantity (p_A is the conditioning number of the matrix A) All estimates are obtained up to terms of higher order of smallness than and ₁. The estimates themselves are related to the classification of errors of computing processes proposed by the author in recent years.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 94–110, 1984.     

Semiclassical Topological Flat Laguerre Planes Obtained by Pasting Along A Circle

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along a circle suitably. Isomorphism classes and automorphism groups of these planes are determined. Together with [9] this gives a complete classification of all semicalssical topological flat Lguerre planes.     

Method for measuring the electron distribution function in the low temperature plasma with a high time resolution

A hybrid method for measuring the electron distribution function in the low temperature plasma is described. The time resolution of the measurement is of the order of several sec.     

Vibrational modes and infrared absorption of interstitial oxygen in silicon

A quasi-linear Si₂O molecule model (QLMM) is suggested from an analysis of the configuration and the interactions of an isolated oxygen atom with its neighbor silicon atoms. The vibrational modes are assigned and the infrared absorption spectra are calculated in detail with the model. The theoretical results are in reasonably good agreement with reported experimental values. This agreement shows that for the analysis of the vibrational modes of the interstitial oxygen atom in silicon crystals it is not necessary to consider the coupling of the molecule with the rest of the lattice. The interaction of the oxygen atom with its six second-nearest silicon atoms only causes the level separation of the ₂ mode and the formation of the fine structure.     

Quantum spin chains: Simple models with complex dynamics

The present study highlights some of the complexities observed in the dynamical properties of one-dimensional quantum spin systems. Exact results for zero-temperature dynamic correlation functions are presented for two contrasting situations:

1. a system with a fully ordered ferromagnetic ground state;
2. a system at aT _c=0 critical point.

For both situations it is found that the exact results are considerably more complex than has been anticipated on the basis of approximate approaches which are considered to be appropriate and reliable for such situations. A still higher degree of complexity is expected for the dynamics of quantum spin systems which are nonintegrable. The paper concludes with some observations concerning nonintegrability effects and quantum chaos in spin systems.