On Some Generalized Symmetric Integral Operators of Buschman-Erdelyi’s Type

Abstract

Some new symmetric integral operators with kernels involving the generalized Legendre’s function of the first kind P ^m,n k (z) are introduced. Some their applications are given.     

Electrostatic field of a thin, unclosed spherical shell and a torus

The electrostatic problem for a thin, unclosed spherical shell and a torus is reduced to paired summation equations in the Legendre polynomials by means of formulas relating the spherical and toroidal harmonic functions. The paired equations are transformed to a Fredholm integral equation of the second kind. Formulas are obtained for computing the charges of the conductors in the form of a series in a small parameter. The capacitance is computed for certain geometrical parameters of the conductors. Zh. Tekh. Fiz. 68, 1–6 (July 1998)     

<Emphasis Type="Italic">SO</Emphasis>(3,1)-Invariant Approach to Dipole Radiation

The aim of the present work is to revisit the theory of the dipole radiation, within an SO(3,1)-gauge invariant formulation, by solving the Maxwell equations. Thus, we obtain the two interconnected components, A _B , B=1,2, of the vector potential A, in terms of Hankel and Legendre polynomials. Finally, for the pure dipole-like radiation, the observables, regarded as phasors, the Umov–Poynting vector components and the well-known Larmor formula for the effective radiated power are explicitly derived.     

Nonlinear moment method for the isotropic Boltzmann equation and invariance of the collision integral

A new approach is proposed for the development of a nonlinear moment method of solving the Boltzmann equation. This approach is based on the principle of invariance of the collision integral with respect to the choice of basis functions. Sonine polynomials with a Maxwellian weighting function are taken as these basis functions for the velocity-isotropic Boltzmann equation. It is shown that for arbitrary interaction cross sections the matrix elements corresponding to the moments of the nonlinear collision integral are not independent but are coupled by simple recurrence formulas by means of which all the nonlinear matrix elements are expressed in terms of linear ones. As a result, a highly efficient numerical scheme is constructed for calculating the nonlinear matrix elements. The proposed approach opens up prospects for calculating relaxation processes at high velocities and also for solving more complex kinetic problems. Zh. Tekh. Fiz. 69, 22–29 (June 1999)     

Spin waves in classical gases

A criterion for the propagation of weakly-damped spin waves is obtained on the basis of a detailed analysis of the quantum collision integral for particles in a paramagnetic polarized Boltzmann gas. It is shown that according to this criterion spin waves can propagate in classical (nonquantum) gases at temperatures close to room temperature. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 10, 777–782 (25 May 1998)     

Isotopic effects of the N(2D) + H2 → NH + H reaction: a quantum time-dependent wave packet investigation

Based on the potential energy surface reported by Li and co-workers (J. Comput. Chem. 34 1686–1696 (2013)), the dynamics calculations of N(²D)?+?H₂(v ₀?=?0, j ₀?=?0) reaction and its isotopic variants HD and D₂ are studied using time-dependent wave packet method in the collision energy range of 0.01–1.0?eV. Dynamics properties such as reaction probability, differential cross section, and integral cross section are studied at state-to-state level of theory. Present values are compared with available theoretical and experimental results. The results indicate that the integral cross sections of N(²D)?+?D₂ reaction are in general good agreement with the experimental data at collision energy below 0.15?eV. The rotational state-resolved integral cross sections of N(²D)?+?H₂/HD/D₂ reactions are compared with experimental values for the first time, with the obtained values being in good agreement with the experimental data.     

Application of Legendre Polynomials Correction Before Component Quantification of Biological Mid-infrared Spectra

Abstract

Quantitative determination from infrared spectra still faces problems that are not completely resolved to date. One of these problems is the deformation of baselines.

We have tested the Legendre baseline correction method by subtracting from a spectrum the low-degreed terms from a Legendre polynomial function which is obtained by the decomposition of the spectrum.

As opposed to the filtered spectra with n=0 which remained very similar to the original spectra, the filtered spectra, as from n=2, exhibit a great amount of distorsion. This result suggests that it is as from n=2, that the terms that are meaningful to the signal (not just to baseline), are to be substracted.

Mathematical methods such as Principal Component Analysis and Principal Component Regression are used to analyse quantitatively components from biological samples. The thus predicted sucrose content values for values of n that range between 0 and 8 (n={0, 2, 4, 6, 8}) show that the best results are obtained for n=4. The mean and standard deviation values of the difference between reference and predicted values varies between 8.5 × 10^?2 and 6.6 × 10^?2 and between 0. 224 and 0. 195 respectively; these values are minimum when the n=4 term is subtracted from the spectra.

This processing hence, sensibly improves the quantification of components from Mid-FTIR spectra of biological solutions. This procedure would be interesting in cases where the precision of quantitative measures is essential.     

Regge pole analysis of large angle elastic scattering in chemically reactive systems

Complex angular momentum techniques are applied to the theory of elastic scattering in chemically reactive systems. The Watson-Sommerfeld transformation allows the scattering amplitude to be written as a background integral and a sum over Regge Poles in the right-hand complex plane. It is shown the Regge Poles are restricted to the first quadrant for physically meaningful optical potentials. Neglect of the background integral and all poles except for the one closest to the real axis allows the differential cross section for large scattering angles to be written in terms of a Legendre function of complex degree. The theory agrees with experimental results for the elastic angular distributions of direct and complex reactions. It predicts oscillations and a glory in the backward direction of the angular distributions. The oscillations arise from interference between orbiting or surface waves that propagate around the region of chemical reaction. Measurements of angular distributions at different collision energies can be used to determine a Regge trajectory.     

Energy relaxation in the spin-polarized disordered electron liquid

Energy relaxation is studied in the spin-polarized disordered electron systems in the diffusive regime. We derive a quantum kinetic equation in which the kernel of the electron-electron collision integral explicitly depends on the electron magnetization. As a consequence, the inelastic scattering rate has a nonmonotonic dependence on the spin polarization of the system.     

Statistical theory of angular momentum polarization in chemical reactions

A general statistical treatment applicable to any vector property of reactive scattering is derived from angular correlation theory. This pertains to the usual experimental situation in which two or three vector directions are observed but numerous other vectors are random or unobserved, particularly various angular momentum vectors. The dependence of the cross section on the angles relating the observed vectors is expanded as a Legendre polynomial series, with coefficients which represent averages of angular momentum functions over the unobserved vectors. An algorithm for calculating these angular correlation coefficients is provided by the statistical theory. All non-vanishing terms involve only even-order Legendre polynomials. In many experiments, one or two terms are predominant. Classical and quantal versions give the same algorithm in the correspondence principle limit, which often holds for chemical reactions. The angular correlations involving the initial and final relative velocity vector directions [kcirc] and [kcirc]′ and the product rotational angular momentum j′ are treated in detail, including both pairwise and triple correlations. Explicit formulae are given for three choices of the quantization axis : along [kcirc], along [kcirc]′, and along [kcirc] × [kcirc]′. Coefficients for the ([kcirc], [kcirc]′, j′) correlations are tabulated for seven reactions as examples and comparison made with recent experimental measurements of the spatial orientation or polarization of j′ in reactions of alkali atoms with hydrogen halides and with methyl iodide.     

Auxiliary Field Functional Integral Representation of the Many-Body Evolution Operator

Finding an appropriate functional integral representation of the many-body evolution operator is a crucial task for performing efficient calculations of fermionic systems within the auxiliary field approach. In this paper we derive a new field representation of the imaginary-time evolution operator using the method of Gaussian equivalent representation of Efimov and Ganbold (1991, Physica Status Solidi 168, 165). The goal is to obtain a functional integral representation, in which the main divergences caused by the tadpole Feynman diagrams are efficiently eliminated. These diagrams provide the main contributions to the ground state of the system under consideration, and therefore it is important to take them into account adequately, especially at lower temperatures. In addition, we show that the well-known mean field representation of the imaginary-time evolution operator is only the limiting case of the Gaussian equivalent representation in the small time-step regime.     

Interference effects in the scattering of identical atoms in a laser radiation field

The properties of the collision integral in a quantum Boltzmann-type kinetic equation are studied under the conditions of spatially nonuniform distributions of colliding particles interacting with an external electromagnetic field. The components of the nonlinear resonances and the velocity distribution of the excited atoms, which are due to polarization transitions, are determined on the basis of the Kazantsev collision integral.     

Path integral approach for Stark broadening of Lyman lines in hydrogen plasma

New results for Lyman lines from hydrogen plasmas are presented using the path integral approach. The influence of plasma components (electrons and ions) on the radiator is analysed separately. The ionic contribution is treated within the path integral approach, while the electronic contribution is estimated by the standard collision operator. The Stark effect, including the ion quadrupole contribution, is considered. The time‐dependent ionic microfield is treated within the path integral approximation using the model microfield method (MMM). The comparison with the quantum statistical approach is performed using a wide range of temperatures (T = 10⁴–10⁷ K) and electron densities (N_e = 10²³–10²⁶ m^?3). Good agreement is mainly obtained for low density and high temperature.     

Expansion of an arbitrary function in products of spherical functions

A formula for expanding an arbitrary function defined in the interval (0,1) into a series in products of spherical Legendre functions of the first and second kinds is presented. Examples of expansions of this type are given. Zh. Tekh. Fiz. 67, 123–126 (June 1997)     

Statistical theory of angular momentum polarization in chemical reactions

A general statistical treatment applicable to any vector property of reactive scattering is derived from angular correlation theory. This pertains to the usual experimental situation in which two or three vector directions are observed but numerous other vectors are random or unobserved, particularly various angular momentum vectors. The dependence of the cross section on the angles relating the observed vectors is expanded as a Legendre polynomial series, with coefficients which represent averages of angular momentum functions over the unobserved vectors. An algorithm for calculating these angular correlation coefficients is provided by the statistical theory. All non-vanishing terms involve only even-order Legendre polynomials. In many experiments, one or two terms are predominant. Classical and quantal versions give the same algorithm in the correspondence principle limit, which often holds for chemical reactions. The angular correlations involving the initial and final relative velocity vector directions [kcirc] and [kcirc]′ and the product rotational angular momentum j′ are treated in detail, including both pairwise and triple correlations. Explicit formulae are given for three choices of the quantization axis: along [kcirc], along [kcirc]′, and along [kcirc] × [kcirc]′. Coefficients for the ([kcirc], [kcirc]′, j′) correlations are tabulated for seven reactions as examples and comparison made with recent experimental measurements of the spatial orientation or polarization of j′ in reactions of alkali atoms with hydrogen halides and with methyl iodide.     

Solution of the transport integral equation with anisotropic scattering

The paper deals with the solution of the integral equation for particle transport in homogeneous material systems having plane and spherical symmetry. Emphasis is put on the explicit inclusion of anisotropic scattering (higher Legendre components of the scattering kernel). The present approach is based on a generalization of the Integral Transform method. The solution is represented as an expansion with respect to analytical basis functions with coefficients satisfying a certain linear system. The determination of this linear system and its matrix elements in a form convenient for numerical purposes is the central point of the paper.     

Short-range order in spin density wave antiferromagnets with chemical dimerization

The spin density wave model in a quasi-one-dimensional itinerant antiferromagnet with staggered potential at finite temperature is studied. Only short-range ordering exists in this system above the Néel temperature. The local-band theory of spin fluctuations is developed to calculate the spin density wave amplitude and the effective exchange integral. The one-electron spectrum and magnon spectrum are obtained in the short-range ordering regime. Zh. éksp. Teor. Fiz. 114, 1346–1364 (October 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Transport properties of a binary mixture of CO2ben N2 from the pair potential energy functions based on a semi-empirical inversion method

The potential energy surface of a CO₂-N₂ mixture is determined by using an inversion method, together with a new collision integral correlation [J. Phys. Chem. Ref. Data 19 1179 (1990)]. With the new invert potential, the transport properties of CO₂-N₂ mixture are presented in a temperature range from 273.15 K to 3273.15 K at low density by employing the Chapman-Enskog scheme and the Wang Chang-Uhlenbeck-de Boer theory, consisting of a viscosity coefficient, a thermal conductivity coefficient, a binary diffusion coefficient, and a thermal diffusion factor. The accuracy of the predicted results is estimated to be 2% for viscosity, 5% for thermal conductivity, and 10% for binary diffusion coefficient.     

20 Ne(34 Ne)原子与18 Na2(23 Na2,37 Na2)分子低温下冷碰撞的同位素效应研究

本文用多体刚性椭球模型计算了相对入射能量为190 meV时,氖的同位素原子²⁰Ne,³⁴Ne与钠的同位素分子¹⁸Na₂,²³Na₂,³⁷Na₂ 替代碰撞体系的态态转动激发积分散射截面和总转动激发积分散射截面,在此基础上计算并分析了相互作用势的不同区域对²⁰Ne-¹⁸Na_{2关键词： 多体刚性椭球模型 转动激发积分散射截面 钠同位素分子 椭球等势面}