Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

Asymptotic behaviour of the Fourier coefficients of MAPW wave functions

In contrast to the conventional APW method the MAPW method imposes continuity conditions on the trial wave functions which guarantee that the wave functions and their first derivatives are continuous throughout the whole atomic polyhedron. It is shown that as a consequence of this fact the Fourier coefficients of the MAPW wave functions approach zero as 1/p ⁴ whereas conventional APW functions decay as 1/p ² only. For the MAPW Fourier coefficients asymptotic formulas are derived that may be useful in numerical calculations. Numerical results are given for metallic Li and Cu.     

半空间球面波函数叠加法重构声源直接辐射声场

提出了基于半空间球面波函数叠加的声场重构方法,以重构含有限声阻抗边界半空间中声源直接辐射的声场.在半空间中多极子声源声压场的解析解的基础上,构造出以边界声阻抗为参量的半空间球面波函数的正交基;通过求逆获得半空间总声压解的基函数系数,同时也获得声源直接辐射声场即自由空间中的基函数系数,进而重构出声源直接辐射的声场.在边界...     

Trilocal structures. III. Expansion in tree functions

An alternative set of expansion functions is described. The first 48 in the set are listed explicitly, and 16 generalized formulas are given, from which the entire rest-system set of functions can be constructed. Explicit and generalized matrix elements for the rest Hamiltonian are given. An auxiliary condition is introduced, leading to explicit formulas and generalized formulas for the expansion coefficients,C _j, accompanying the expansion functions, _j.     

二维高斯波束对多层球粒子电磁散射的解析解

基于矢量波函数在球和柱坐标系中表达式之间的转换关系,提出了一种求解球坐标系中二维高斯波束波形因子的方法,得到了二维高斯波束波形因子在球坐标系中的解析公式.结合广义米理论推导了在轴二维高斯波束入射多层球粒子的电磁散射的解析解,并对散射强度随散射角的分布进行了数值模拟,结果与平面波入射情况进行了比较. 关键词： 矢量波函数 波形因子 电磁散射 广义米理论     

Exact nonrelativistic expressions for the tensor for scattering of light by atoms

Exact nonrelativistic analytical expressions are derived for dipole two-photon transitions between arbitrary multiplets of the hydrogen atom and positive hydrogenlike ions. The result is expressed in terms of a single Gauss hypergeometric function and polynomials whose degrees increase linearly with the number of nodes of the bound states of the quantum system. The cross sections of elastic scattering of light by K-and L-shells of the hydrogen atom are given as an example. It is demonstrated that by expanding the discrete-spectrum wave functions in ultraspherical polynomials it is also possible to obtain analytical expressions of the cross sections of two-photon transitions between states described by the Simons model potential. The basis consisting of Chebyshev polynomials is shown to be the best expansion basis, and the coefficients of such an expansion are given for a broad range of parameters of the problem. Calculation of the polarizability of the 5S-state of the rubidium atom is chosen as an example. Finally, the results are compared with the experimental data and the theoretical results of other researchers. Zh. éksp. Teor. Fiz. 111, 816–830 (March 1997)     

群表示论的物理方法(Ⅴ)——SU3?SO3和SU4?SU2×SU2分类基

本文给出了核物理中常用的SU₃?SO₃(f=1—6)和SU₄?SU₂×SU₂波函数在Gelfand表象中的展开系数。 关键词：     

Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.     

Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension

In this paper we compute the coefficients of the heat kernel asymptotic expansion for Laplace operators acting on scalar functions defined on the so called spherical suspension (or Riemann cap) subjected to Dirichlet boundary conditions. By utilizing a contour integral representation of the spectral zeta function for the Laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure. Thanks to the well known relation between the zeta function and the heat kernel obtainable via Mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions. The particular case of a d-dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given.     

Diffraction of a Plane Electromagnetic Wave by an Infinite Double-Ridge Wedge with a Dielectric Cylinder at the Crest

We reduce the considered problem to solving a matrix equation of the second kind for unknown coefficients of expansion of a diffracted field into a Fourier–Bessel series. This expansion was obtained by imposing boundary conditions on the diffracted field with the subsequent re-expansion of the field function over basis functions in a given interval. The expansion coefficients were determined analytically in the case where the electric diameter of the cylinder is less than unity as well as numerically with a high accuracy by solving the obtained matrix equation using the reduction method. We derived expressions for the pattern of the far-zone field scattered by the studied structure and the backscattering cross section and give exact numerical results for the case of an E-polarized incident wave.     

Transformation Coefficients of Hyperspherical Harmonic Functions of an -Body System

Two algorithms are presented for calculating the transformation coefficients between hyperspherical harmonic functions constructed with different sets of Jacobi vectors. They have been tested in the case , where the transformation coefficients of states with grand angular quantum number up to have been studied. The applicability of the two algorithms to larger systems is discussed. The numbers of independent hyperspherical-spin-isospin states with given values, entering the expansion of the alpha-particle ground-state wave function, are also evaluated. The use of complete non-redundant bases is important for future accurate applications of the hyperspherical harmonic technique. Received December 23, 1997; revised May 25, 1998; accepted for publication May 30, 1998     

Multipole expansion of non-local coupling potentials for cluster transfer and application to 16O(16O, 12C)20Ne

When the transfer of clusters and the symmetrization (antisymmetrization) of scattering wave functions is described by cluster models within the coupled-channel formalism, non-local coupling potentials arise. We suggest a procedure to calculate these potentials by a multipole expansion of all potentials and wave functions which depend on sums of vectors. The expansion coefficients are found by least-squares fit. The method is applied to the ¹⁶O(¹⁶O, ¹²C)²⁰Ne reaction, which is treated in the cluster model with two inert ¹²C- and α-clusters as constituents.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理（HVT）和Hellmann-Feynman定理（HFT）,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出由能级近似值计算定态波函数近似表达式的代数公式,给出了代数公式具体应用的几个典型一维势阱实例,此法可推广到二维势阱与三维势阱的情形。     

General multipole expansion of polarization observables in deuteron electrodisintegration

Formal expressions are derived for the multipole expansion of the structure functions of a general polarization observable of exclusive electrodisintegration of the deuteron using a longitudinally polarized beam and/or an oriented target. This allows one to exhibit explicitly the angular dependence of the structure functions by expanding them in terms of the small rotation matrices d ^j _m'm(θ), whose coefficients are given in terms of the electromagnetic multipole matrix elements. Furthermore, explicit expressions for the coefficients of the angular distributions of the differential cross-section including multipoles up to L _max = 3 are listed in tabular form. Received: 19 November 2002 / Accepted: 7 May 2002     

Exact partial wave expansion of optical beams with respect to an arbitrary origin

Using an analytical expression for an integral involving Bessel and Legendre functions, we succeed in obtaining the partial wave decomposition of a general optical beam at an arbitrary location relative to the origin. We also showed that solid angle integration will eliminate the radial dependence of the expansion coefficients. The beam shape coefficients obtained are given by an exact expression in terms of single or double integrals. These integrals can be evaluated numerically on a short time scale. We present the results for the case of a linear-polarized Gaussian beam.     

Coordinate representation of the McWeeny-Coulson wave function for the helium atom

The coordinate representation of the McWeeny-Coulson wave function for the helium atom is obtained in the form of an expansion in scalar tripolar spherical harmonics. The coefficients of the expansion, which are functions of the interparticle coordinates, are represented as a sum of terms admitting separation of variables. Analysis of the expansion constructed substantiates the presence of the Huelten correlation factor in the wave function as well as the appearance of Huelten orbitals.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 37–43, August, 1985.     

Acoustic scattering by a penetrable spheroid

The scattering of a plane acoustic wave from a penetrable prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the various scattering cross-sections when the solution is specialized to small values of the eccentricity h = d/(2a), (h ≪ 1), with d being the interfocal distance of the spheroid and 2a the length of its rotation axis. In this case, exact, closed-form expressions are obtained for the expansion coefficients g ⁽²⁾ and g ⁽⁴⁾ in the relation S(h) = S(0)[1 + g ⁽²⁾ h ² + g ⁽⁴⁾ h ⁴ + O(h ⁶)] expressing the scattered field and the scattering cross-sections. Numerical results are given for various values of the parameters. Published in Russian in Akusticheskiĭ Zhurnal, 2008, Vol. 54, No. 2, pp. 189–204. The text was submitted by the authors in English.     

Hadronic quark distribution amplitudes from QCD sum-rule moments

We discuss the reliability of hadronic wave functions (quark distribution amplitudes) determined by a finite number of QCD sum-rule moments. Although the expansion coefficients for polynomial models of the wave function are uniquely determined by the moments, the inherent uncertainty in such moments leads to a considerable indeterminacy in the wave functions because minimal changes of the moments can lead to large oscillations of the model function. In particular, the freedom in the moments left by QCD sum rules leads to a nonconverging polynomial expansion. This remains true even if additional constraints on the wave functions are used. As a consequence of this, the widely used procedure of constructing polynomial models of hadronic wave function from QCD sum rule moments does not guarantee even a reasonable approximation to the true wave function. The differences among the model wave functions persist also in the calculations of physical observables like hadronic form factors. This implies that physical observables calculated by means of such model wave functions are in general very unreliable. As specific examples, we examine the pion and nucleon wave functions and show that Gegenbauer as well as Appell polynomial expansions constructed from QCD sum rule moments are ruled out. The implications for the wave functions which are generally used in the literature are discussed.     

Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation

Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

Acoustic scattering by an impenetrable spheroid

The scattering of a plane acoustic wave from an impenetrable, soft or hard, prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the various scattering cross-sections, when the solution is specialized to small values of the eccentricity h = d/(2a) , where d is the interfocal distance of the spheroid and 2a is the length of its rotation axis. In this case, exact, closed-form expressions are obtained for the expansion coefficients g ⁽²⁾ and g ⁽⁴⁾ in the relation S(h) = S(0)[1 + g ⁽²⁾ h ² + g ⁽⁴⁾ h ⁴ + O(h ⁶)] expressing the scattered field and the scattering cross-sections. Numerical results are given for various values of the parameters. Published in Russian in Akusticheskiĭ Zhurnal, 2007, Vol. 53, No. 4, pp. 500–513. The text was submitted by the authors in English.