Asymptotic fluctuations of a particle system with singular interaction

A particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force is considered. The empirical measure process is known to converge weakly in a space of continuous-measure-valued functions. In this work we show that the fluctuations around the limiting process, appropriately scaled, converge weakly to a Gaussian-distribution-valued process.     

Completely integrable model of classical field theory with nontrivial particle interaction in two-dimensional space-time

The complete integrability of a nonlinear relativistic equation, being a generalization of the sine-Gordon equation for the case of group SO(3), is shown. The soliton spectrum is described and exact solutions describing their interaction are obtained. It is shown that together with elastic scattering there exist the processes of decay and merging of solitons. A generalization of the sine-Gordon equation for an arbitrary compact Lie group is discussed.     

Vacuum particle creation under action of a strong external field: an example of irreversible behavior of a system with time reversal symmetry

On the basis of a kinetic equation (KE) of the Vlasov type, which is a strong non-perturbative consequence of the QED equations of motion in the quasiparticle representation, we investigate the irreversible behavior of the electron-positron plasma (EPP) created from vacuum under action of a pulsed external quasiclassical field. The basic KE is upon time reversal. Nevertheless the EPP reveals some features characteristic for irreversible processes. The nondecreasing entropy principle is introduced for the asymptotic in- and out-states: S _out ?? S _in . Some features of the entropy evolution are investigated for different excitation levels of the EPP.     

Analytic dynamics of a one-dimensional system of particles with strong interaction

We consider the dynamics of a system of N particles on the circle with interaction of nearest neighbors, a Coulomb potential, and an analytic external force. The trajectories are real analytic functions of time. However, the series for them converge only for sufficiently small times. For zero initial velocities and a uniform initial location of particles, we prove N-dependent estimates on the coefficients of this series.     

Quantum system level-band in a strong quasiresonant field with a deep modulation

The dynamics of a level-band quantum system in a strong external quasiresonant laser field with a deep modulation is discussed. It is assumed that the amplitude of the external field considerably exceeds the width of the band. The related Schrödinger equation is reduced to a singular integro-differential equation with a large parameter. Using appropriate asymptotic techniques, the decay of the level population is examined. Bibliography: 12 titles.     

Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field,perturbed by a decreasing electric field. I

One investigates the bound statesE _n , situated to the left of the boundary of the continuous spectrum. Under the assumption that the potential is nonpositive and has at infinity a power type asymptotics, one computes the principal term of the asymptoticsE _n as n. One does not assume the axial symmetry of the potential of the electric field.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 67–84, 1984.In conclusion, the author expresses his deep gratitude to his scientific adviser M. Sh. Birman for the formulation of the problem and for his constant interest in the paper.     

The limiting Lagrangian as a consequence of Helly's theorem

The perturbational Lagrangian equation established by Jeroslow in convex semi-infinite programming is derived from Helly's theorem and some prior results on one-dimensional perturbations of convex programs.This research was partially supported by NRC, Grant No. A-4493.     

Long-time asymptotics for a classical particle interacting with a scalar wave field

We consider the Harniltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set of stationary solutions in the long time limit t f oo. The rate of relaxation to a stable stationary solution is determined by spatial decay of initial data. 'Supported partly by French-Russian A.M.Liapunov Center of Moscow State University, by research grants of RFBR (9601-00527) and of Volkswagen-Stiftung.     

Nonequilibrium state of a ferroelectric in a strong magnetic field

Vitebsk Branch of the Institute of Solid-State Physics and Semiconductors, Belorussian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 1, pp. 147–153, January, 1989.