To the Jubilee of Vsevolod Alekseevich Solonnikov

Bibliography: 42 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 7–15.     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

Nonequilibrium Ionization by Vibrationally Excited Molecules in Supersonic Flows

The results of experimental investigations on thermal nonequilibrium ionization in CO₂: N₂: He mixtures are presented. Measurements of electron density, n_e, in vibrationally excited nitrogen were made in a supersonic flow with different CO₂ contents as well as in a CO₂: N₂: He = 1 : 5 : 4 mixture laser gas. The mixtures were heated in a shock tube and expanded through a supersonic nozzle. Furthermore, supersonic mixing of N₂ and CO₂ + He was used in some experiments. The measured values of n_e in the plenum chamber and in the supersonic nozzle are reported, and the processes responsible for nonequilibrium ionization in a laser-active medium are discussed.     

Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.     

Lattice vibrations of armchair carbon nanotubes: phonons,soliton deformations and lattice discreteness effects

Small and large-amplitude elastic deformations of the armchair structure of single-walled carbon nanotubes are investigated with emphasis on the cylindrical geometry. As starting model, we consider a discrete one-dimensional lattice of atoms interacting via a Lennard-Jones type two-body potential. In an expansion scheme using cylindrical coordinates where radial displacements are assumed negligible compared to the angular motions, a sine-lattice Hamiltonian is derived. In the limit of small-amplitude angular displacements, the dispersion spectrum of acoustic phonons is derived and the associate characteristic frequency is given as a function of parameters of the model. In the large-amplitude regime, lattice vibrations give rise to kink-type deformations which move undergoing lattice dispersion and lattice discreteness effects. The dispersion law of the kink motion is obtained and shown to lower the effect of lattice discreteness, giving rise to a vanishing Peierls stress for kink sizes of the order of a few lattice spacings. Implications of the coupling of two armchair structures on the stability of vibrational modes of an individual armchair nanotube are also discussed. A gap of forbidden modes is predicted in the phonon spectrum while the energy needed to create a kink deformation in individual nanotubes is shifted in the presence of a wall-to-wall interaction.Received: 2 August 2004, Published online: 14 December 2004PACS: 81.07.De Nanotubes - 62.30. + d Mechanical and elastic waves-vibrations - 63.22. + m Phonons in low-dimensional nanoscale materials - 63.20.Ry Anharmonic lattices modes     

Backscattering in the vicinity of a mode cutoff

In a recent paper to this journal (Whitman A M et al 2003 Waves Random Media 13 269-86) we derived a set of coupled equations that describe the intermodal scattering of acoustic radiation in a duct whose speed of sound varies randomly in space and time. In the paper we were mainly interested in modes that were not near cutoff. Here we study the solution of these equations in the vicinity of the cutoff. We find that near cutoff almost all the energy is reflected back independent of the other duct parameters. In addition to presenting these results, we analyse the mathematical structure of the equations in these regions in order to elucidate the reason for this behaviour.     

Rate constants for the gas‐phase reaction of CF3CF2CF2CF2CF2CHF2 with OH radicals at 250–430 K

The rate constants k₁ for the reaction of CF₃CF₂CF₂CF₂CF₂CHF₂ with OH radicals were determined by using both absolute and relative rate methods. The absolute rate constants were measured at 250–430 K using the flash photolysis–laser‐induced fluorescence (FP‐LIF) technique and the laser photolysis–laser‐induced fluorescence (LP‐LIF) technique to monitor the OH radical concentration. The relative rate constants were measured at 253–328 K in an 11.5‐dm³ reaction chamber with either CHF₂Cl or CH₂FCF₃ as a reference compound. OH radicals were produced by UV photolysis of an O₃–H₂O–He mixture at an initial pressure of 200 Torr. Ozone was continuously introduced into the reaction chamber during the UV irradiation. The k₁ (298 K) values determined by the absolute method were (1.69 ± 0.07) × 10^?15 cm³ molecule^?1 s^?1 (FP‐LIF method) and (1.72 ± 0.07) × 10^?15 cm³ molecule^?1 s^?1 (LP‐LIF method), whereas the K₁ (298 K) values determined by the relative method were (1.87 ± 0.11) × 10^?15 cm³ molecule^?1 s^?1 (CHF₂Cl reference) and (2.12 ± 0.11) × 10^?15 cm³ molecule^?1 s^?1 (CH₂FCF₃ reference). These data are in agreement with each other within the estimated experimental uncertainties. The Arrhenius rate constant determined from the kinetic data was K₁ = (4.71 ± 0.94) × 10^?13 exp[?(1630 ± 80)/T] cm³ molecule^?1 s^?1. Using kinetic data for the reaction of tropospheric CH₃CCl₃ with OH radicals [k₁ (272 K) = 6.0 × 10^?15 cm³ molecule^?1 s^?1, tropospheric lifetime of CH₃CCl₃ = 6.0 years], we estimated the tropospheric lifetime of CF₃CF₂CF₂CF₂CF₂CHF₂ through reaction with OH radicals to be 31 years. © 2003 Wiley Periodicals, Inc. Int J Chem Kinet 36: 26–33, 2004