On some integrable systems in the extended lobachevsky space

Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schrödinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.     

Bogoliubov transformation and Lobachevsky planimetry

The relation between the Bogoliubov transformation in the theory of superfluidity and the Lobachevsky planimetry in two conformally Euclidean Poincare models is established. The parameter of this transformation, a complex number with the modulus <1, is represented as a radius vector on the Lobachevsky plane.     

Eigenfunction expansions in the imaginary Lobachevsky space

Eigenfunctions of the Schrödinger equation with the Coulomb potential in the imaginary Lobachevsky space are studied in two coordinate systems admitting solutions in terms of hypergeometric functions. Normalization and coefficients of mutual expansions for some sets of solutions are found.     

Lobachevsky geometry and unsolved problems of solar cosmogony

An axiomatic method of constructing physics in Lobachevsky space is proposed. Dynamical equations are applied to explain the principal properties of the solar system, its planets, their satellites, and rings.     

Magnetic field in the Lobachevsky space and related integrable systems

Various possibilities to define analogs of the uniform magnetic field in the Lobachevsky space are considered using different coordinate systems in this space. Quantum mechanical problem of motion in the defined fields is also treated. Variables in the Schr?dinger equation are separated and diagonal operators are found. For some cases, exact solutions are obtained.     

On models of the visual space

Linear and nonlinear models of the visual space arising in human consciousness upon visual perception of the surrounding world are proposed. The models correspond to monocular vision and satisfy Emmert’s psychophysical law. The relative dimensions of images of objects obtained according to the laws of geometrical optics and perceptive perspective are compared with known experimental data. The nonlinear model is found to coincide with the Klein model of the Lobachevsky space.     

Generating function for extended Jacobi polynomials, noncommutative differential calculus and the relativistic energy and momentum operators

It is shown that the generating function for the matrix elements of irreps of Lorentz group is the common eigenfunction of the interior derivatives of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions. These derivatives commute and can be interpreted as the quantum mechanical operators of the relativistic momentum corresponding to the half of the non-Euclidean distance from the origin in the Lobachevsky momentum space.     

Interactions of aj=1 boson in the 2(2j+1) component theory

The amplitudes for boson-boson and fermion-boson interactions are calculated in the second order of perturbation theory in the Lobachevsky space. An essential ingredient of the model is the Weinberg 2(2j+1)-component formalism for describing a particle of spinj. The boson-boson amplitude is then compared with the two-fermion amplitude obtained long ago by Skachkov on the basis of the Hamiltonian formulation of quantum field theory on the mass hyperboloid,p ₀ ² –P ²=M ², proposed by Kadyshevsky. The parametrization of the amplitudes by means of the momentum transfer in the Lobachevsky space leads to the same spin structures in the expressions ofT matrices for the fermion and the boson cases. However, certain differences are found. Possible physical applications are discussed.On leave of absence from Department of Theoretical and Nuclear Physics, Saratov State University, Saratov, Russia. E-mail: dvoeglazov@mainl.jinr.dubna.su.     

Lobachevsky geometry of (super)conformal mechanics

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) . We also extend this picture to the N=2k superconformal mechanics described in terms of Lobachevsky superspace .     

Estimation of the global stability domain in a chain of correlated phase-locked systems by the methods of image recognition theory

We formulate the problem and present the results of estimation of the global stability domain of the equilibrium state in a chain of correlated phase-locked systems by the methods of image recognition theory. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Lobachevsky State University of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1612–1615, December, 1998.     

Effect of an RF discharge plasma on the telemetric antenna radiation of a sounding rocket

A considerable increase in the intensity of the telemetric signal radiated from the MR 12 sounding rocket surrounded by an artificial RF discharge plasma envelope was detected in the “Aktivny Shnur” ionospheric experiment in February 1991. The experimental data are presented and their interpretation is given. Lobachevsky State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 1, pp. 121–126, January, 1998.     

A hundred years of special theory of relativity

A special theory of relativity is considered here as an episode of non-Euclidean geometry. Special attention is drawn to the fact that the replacement of the fifth Euclidean postulate by the Lobachevsky postulate of parallel straight lines in the space of velocities of a material point leads to the replacement of the postulate of the same time rate by the postulate of the same velocity of light in all inertial reference systems.     

Relativistic kinetic momentum operators

In quantum theory, in the relativistic configuration r-space, the kinetic momenta, corresponding to the half of the non-Euclidean distance in the Lobachevsky velocity space, are introduced. These operators, coinciding up to the constant factor with the generators of translations of the r-space, are the exterior derivatives of noncommutative differential calculus.     

Point perturbations in constant curvature spaces

Point perturbations of the free Hamiltonian in two- and three-dimensional spaces of constant curvatures are considered. The study of the spectral properties of perturbed Hamiltonian and various asymptotics for its point levels are presented. It is shown that the binding energy in comparison with the case of zero curvature reduces in the case of Lobachevsky plane and rises in the case of 2D-sphere, when the scattering length is much less than the curvature radius.     

Linear prediction based on the solution of an eigenvalue problem for the autocorrelation matrix

To solve the widespread data-processing problem of predicting the signal or process behavior, we propose a unconventional method for determining the prediction coefficients, which is based on the solution of the problem of finding the eigenvalues of the autocorrelation matrix of the process. The method has some advantages. In particular, the roots of the characteristic polynomial based on the prediction coefficients are located on the unit circle. Physical and Technical Research Institute, N. I. Lobachevsky State University, Nizhny Novgorod, Russia Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 1, pp. 66–70, September 2000.     

Relativistic kinetic momentum operators, half-rapidities and noncommutative differential calculus

It is shown that the generating function for the matrix elements of irreps of Lorentz group is the common eigenfunction of the interior derivatives of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions in the Relativistic Configuration Space (RCS). These derivatives commute and can be interpreted as the quantum mechanical operators of the relativistic momentum corresponding to the half of the non-Euclidean distance in the Lobachevsky momentum space (the mass shell).     

New mechanism of Brownian transport

We propose a new model of transport of Brownian particles by deterministic modulation of the ratchet potential. Such a mechanism results in a directed uniform drift of Brownian particles rather than in their diffusion. Lobachevsky State University of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 1, pp. 87–91, January, 1999.     

Optimal estimation of the parameters of a nonstationary random pulse train in discrete time

The problem of optimal estimation of the parameters of a nonstationary sequence of random pulses observed additively with the noise is considered with the step change in average pulse repetition rate taken into account. A recurrent algorithm is derived for optimal estimation of the pulse repetition rate and time of its jump in real time. The optimal assessment of the Bernoulli trial parameters is shown as an example. The results of computer simulation experiments are presented.Lobachevsky State University, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 7, pp. 678–694, July, 1995.     

Density and particle concentration of a passive impurity in the burgers velocity field

An exact solution of the equation for the impurity concentration is found in the case where the density field of the medium is potential and is described by the Burgers multidimensional equation in the space of arbitrary dimension N. The behavior of the active impurity concentration in three-dimensional space is analyzed and the dependences of the generalized density in the one-dimensional case as applied to the evolution of the large-scale distribution of the matter structure in the Universe are studied. Lobachevsky State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 9, pp. 1144–1154, September, 1997.     

Detecting and filtering a flow of different pulsed signals against the noise background

We consider the problem of separate detection and filtration of a flow of pulsed signals. We assume that statistically independent pulsed signals of different shape and random amplitude appear randomly against the background of noise. An algorithm for estimation of the detection probability of each signal is obtained. Recurrent expressions for thea posteriori probability density of each signal are found. The results of numerical simulation of the proposed algorithms of signal processing are presented. Lobachevsky State University of Nizhny Novgorod Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 1, pp. 92–100, January, 1999.