Scattering of charged particles by a multicenter potential

Exact expressions are obtained for the amplitude and elastic cross section in the case of scattering of charged particles by a multicenter pseudopotential that includes the Coulomb potential and an arbitrary number of short-range potentials (modeled by zero-range potentials). Asymptotic limits are calculated and explicit expressions are found for the amplitudes of scattering by few-nucleon complexes modeled by superpositions of the Coulomb potential and purely point potentials.N. N. Bogolyubov Institute of Theoretical Physics, Ukranian Academy of Sciences; T. Shevchenko Kiev University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 1, pp. 81–91, April, 1994.     

Stability of phonon branch of the Bose spectrum of a superfluid Fermi system for large momenta

The phonon branch of Bose spectrum of a superfluid Fermi system (Bogolyubov sound) is investigated atT=0. It is shown that a Bogolyubov sound quantum (with arbitrary momentum) is stable and cannot decay into several excitations with lower energy.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 114–117, 1983.     

Molecular hydrodynamics of a weakly degenerate nonideal bose-gas II. Green functions and kinetic coefficients

The nonideal degenerate Bose-system at a temperature close to zero is investigated by N. N. Bogolyubov and D. N. Zubarev's method of collective variables. Applicable to a wide range of frequencies and wave vectors, interpolated expressions for the Green functions of the densities of conserved quantities and for the superfluid velocity potential are derived from exact relations. The criteria defining the validity domain of the hydrodynamic approximation are obtained. The kinetic coefficients of two-liquid hydrodynamics at temperatures close to zero are calculated for a weakly nonideal Bose-gas. A comparison with the method employing the kinetic equation for quasi-particles is made.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 1, pp. 77–129, October, 1995.     

On a modification of the averaging method for seeking higher approximations

Systems in the N.M. Bogolyubov standard form as well as systems with rapid phases are considered. It is proposed to seek the solution in the form of an asymptotic series in a small parameter with coefficients representable in the form of the sum of two functions. The first depends on slow time and is found as the solution of a simpler equation in a finite segment. The second is a trigonometric polynomial of the time (or the angular displacements) with coefficients which depend on the slow time (it is found in an explicit manner). It is convenient to use the results in solving certain problems in celestial mechanics.     

The stability of some classes of systems in standard Bogolyubov form

The stability of the equilibrium of quasilinear systems in standard Bogolyubov form is investigated. Classes of systems are distinguished out for which it is possible to determine the threshold value ?₀ of the small parameter ? which ensures qualitative agreement between the solutions of the initial systems of equations and the solutions of the averaged system corresponding it in an infinite time interval when ? < ?₀.     

Gamkrelidze Convexification and Bogolyubov’s Theorem

Mathematical Notes - The interconnection between an optimal control problem and its convexification in the sense of Gamkrelidze and Bogolyubov is studied. Bogolyubov’s classical result for...     

Averaging of differential inclusions

We prove a version of the Krasnosel’skii-Krein theorem for differential inclusions with multimappings satisfying certain one-sided constraints. As a corollary, we obtain an analog of the first Bogolyubov theorem for the inclusion 0 ? x′ + ??(t, x).     

Spectrum and states of the BCS Hamiltonian with sources

We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the thermodynamic limit. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1243–1269, September, 2008.     

Existence Theorems for the Initial Value Problem for the Bogolyubov Chain of Equations in the Space of Sequences of Bounded Functions

We prove that the Cauchy problem for a nonsymmetric Bogolyubov chain of equations has a solution representable as an expansion in particle groups (clusters) whose evolution is governed by the cumulant (semi-invariant) of the evolution operator for this particle group in the space of sequences of summable and bounded functions.     

On the theory of high-temperature superconductivity

The construction of a theory of the superconductivity of oxide metals based on the polar model is discussed. The matrix electron Green's function is calculated with allowance for both the electron—phonon interaction and strong Coulomb correlations. The possible raising ofT _c due to the lattice instability associated with strong anharmonicity of the oxygen ion vibrations in perovskitelike compounds is considered. Strong electron correlations are taken into account in a one-bandt—J model. Because of a local restriction that precludes doubly occupied states in a system with strong correlation, onlyd-wave pairing is possible.N. N. Bogolyubov (deceased)The present paper was presented at the Fifth International Symposium on Selected Problems of Statistical Mechanics (August 22–24, 1989, Dubna) and is published here with small additions made without the participation of N. N. Bogolyubov.Joint Institute for Nuclear Research, 141980, Dubna, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 371–383, December, 1992.     

BCS Model Hamiltonian of the Theory of Superconductivity as a Quadratic Form

Bogolyubov proved that the average energies (per unit volume) of the ground states for the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide in the thermodynamic limit. In the present paper, we show that this result is also true for all excited states. We also establish that, in the thermodynamic limit, the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide as quadratic forms.     

Periodic solutions of parabolic inclusions and the averaging method

We obtain an analog of the second Bogolyubov theorem for differential inclusions with multimappings acting in Sobolev spaces and satisfying some monotonicity and compactness conditions. As a consequence, we obtain criteria for the existence of periodic solutions of secondorder parabolic inclusions.     

Asymptotics of the probability of values of random Boolean expressions

The random Boolean expressions are considered that are obtained by the random and independent substitution with the probabilities p and 1 ? p of the constantly one function and constantly zero function for variables of repetition-free formulas over a given basis. The probability is studied that the expressions are equal to one. It is shown that, for each finite basis and p ? (0, 1), this probability tends to some finite limit P ₁(p) as the length of an expression grows. Explicit representation of the probability function P ₁(p) is found for all finite bases, the analytic properties of this function are studied, and its behavior is investigated in dependence on the properties of the basis.     

Second bogolyubov theorem for systems of difference equations

We establish an analog of the second Bogolyubov theorem for a system of difference equations.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — asymptotic decomposition. We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1171–1188, September, 1994.     

The averaging method and the asymptotic behavior of solutions to differential inclusions

We prove one version of the first Bogolyubov theorem for differential inclusions with multivalued mappings that satisfy certain one-sided constraints. We study the dependence of solutions to differential inclusions on the parameters.     

Approximate solution of the Fokker-Planck-Kolmogorov equation

For the Fokker-Planck-Kolmogorov equation, the higher approximations are constructed by using the Bogolyubov averaging method.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 351–361, March, 1995.     

Zitterbewegung of a band electron

We should like to dedicate this paper to the remarkable anniversary in the life of Academician N. N. Bogolyubov, whose outstanding studies always stimulated our modest investigations, including the present paper.     

Normalization and averaging on compact lie groups in nonlinear mechanics

We consider the method of normal forms, the Bogolyubov averaging method, and the method of asymptotic decomposition proposed by Yu. A. Mitropol’skii and the author of this paper. Under certain assumptions about group-theoretic properties of a system of zero approximation, the results obtained by the method of asymptotic decomposition coincide with the results obtained by the method of normal forms or the Bogolyubov averaging method. We develop a new algorithm of asymptotic decomposition by a part of the variables and its partial case — the algorithm of averaging on a compact Lie group. For the first time, it became possible to consider asymptotic expansions of solutions of differential equations on noncommutative compact groups.     

Relaxation in the optimal control problem for the Goursat-Darboux system

We consider the optimal control problem for a system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls satisfying mixed nonconvex constraints. For this problem we prove an analog of the classical Bogolyubov relaxation theorem.