Hyperspherical Functions Research Articles

Study of the Structure of the 9Be Nucleus in the Alpha-Cluster Model by the Method of Hyperspherical Functions

Study of the Structure of the 9Be Nucleus in the Alpha-Cluster Model by the Method of Hyperspherical Functions

Antisymmetrization of the wave functions consisting of spin-isospin and hyperspherical parts

In order to investigate few-particle hypernuclei within the framework of the Hyperspherical Function Method, it is necessary to obtain full wave functions that are anti-symmetric under particle interchange. These wave functions must include not only hyperspherical, but also spin-isospin parts. According to the Parentage Scheme of Symmetrization, N-particle hyperspherical functions (N=3,4,5,6…) symmetrized with respect to (N-1)-particles, can be obtained from the N-particle hyperspherical functions with arbitrary quantum numbers by the use of the transformation coefficients related with the permutations of the last two particles. This article explains how to obtain fully antisymmetrized wave functions consisting of spin-isospin and hyperspherical parts. It is demonstrated that there are sixteen possible combinations in (3+1) configuration and 12 possible combinations in (2+2) configuration when spin and isospin functions are represented by [4], [31], and [22] representations of the four-particle permutation group S4. A complete set of the fully antisymmetrized four-particle wave functions is obtained. It is demonstrated that proposed mathematical formalism can be easily generalized to obtain fully antisymmetrized wave functions for the systems of five and more particles.

Study of Cluster Structure of Light Nuclei by Feynman’s Continual Integrals and Hyperspherical Functions

Calculations of the probability densities and energies for Nα-cluster nuclei 12C (3α), 16O (4α), 20Ne (5α), 24Mg (6α), 28Si (7α) and for nuclear molecules 9Be (α + n + α), 10Be (α + n + n + α), were performed by the Feynman’s continual integrals method using parallel computing based on NVIDIA CUDA technology. New effective method for the solution of the hyperradial equations with cubic splines approximation is proposed and tested for exactly solvable 4-body system.

Computing an orthonormal basis of symmetric or antisymmetric hyperspherical harmonics

A numerical method to build an orthonormal basis of properly symmetrized hyperspherical harmonic functions is developed. As a part of it, refined algorithms for calculating the transformation coefficients between hyperspherical harmonics constructed from different sets of Jacobi vectors are derived and discussed. Moreover, an algorithm to directly determine the numbers of independent symmetric hyperspherical states (in case of bosonic systems) and antisymmetric hyperspherical-spin–isospin states (in case of fermionic systems) entering the expansion of the A-body wave functions is presented. Numerical implementations for systems made with up to five bodies are reported.

Computation of Hyperspherical Bessel Functions

AbstractIn this paper we present a fast and accurate numerical algorithm for the computation of hyperspherical Bessel functions of large order and real arguments. For the hyperspherical Bessel functions of closed type, no stable algorithm existed so far due to the lack of a backwards recurrence. We solved this problem by establishing a relation to Gegenbauer polynomials. All our algorithms are written in C and are publicly available at Github [https://github.com/lesgourg/class_public]. A Python wrapper is available upon request.

Raynal–Revai coefficients for a general kinematic rotation

In a three-body system, transitions between different sets of normalized Jacobi coordinates are described as general kinematic transformations that include an orthogonal or a pseudoorthogonal rotation. For such rotations, the Raynal–Revai coefficients execute a unitary transformation between three-body hyperspherical functions. Recurrence relations that make it possible to calculate the Raynal–Revai coefficients for arbitrary angular momenta are derived on the basis of linearized representations of products of hyperspherical functions.

A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters

In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.

Lightest Kaonic Nuclear Clust

We present our study of kaonic three-body KNN , KNN and KKK and four-body KNNN , and KKNN clusters within the framework of a potential model using the method of hyperspherical functions in momentum representation. To perform a numerical calculations for the bound state energy of the light kaonic system, we use a set of different potentials for the nucleon-nucleon and KN interactions, as well as for the kaon-kaon interaction. The calculations show that a quasibound state energy is not sensitive to the NN interaction, and it shows very strong dependence on the KN potential. We also compare our results with those obtained using different theoretical approaches. The theoretical discrepancies in the binding energy and width for the lightest kaonic system related to the different NN and KN interactions are addressed.

Convergent close coupling versus the generalized Sturmian function approach: Wave-function analysis

Fil: Ambrosio, Marcelo Jose. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Astronomia y Fisica del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomia y Fisica del Espacio; Argentina

Multi-dimensional extreme learning machine

As an important branch of neural network, extreme learning machines (ELMs) have attracted wide interests in the fields of pattern classification and regression estimation. However, when facing learning problems with multi-dimensional outputs, named multi-dimensional regression, the conventional ELMs could not generally get satisfactory results because it is incapable of exploiting the relatedness among outputs efficiently. To solve this problem, a new regularized ELM is firstly proposed in this paper by introducing a hyper-spherical loss function as regularizer. As the regularization form with this loss function cannot be solved directly, an solution with iterative procedure is presented. For improving the learning performance, the algorithm proposed above is further reformulated to identify the inner grouping structure hidden in outputs by assuming that the grouping structure is determined by different linear combinations of a small number of latent basis neurons. This is achieved as a mixed integer programming, and finally an alternating minimization method is presented to solve this problem. Experiments on two multi-dimensional data sets, a toy problem and a real-life dynamical cylindrical vibration data set, are conducted, and the results demonstrate the effectiveness of the proposed algorithm.

Fast and accurate CMB computations in non-flat FLRW universes

We present a new method for calculating CMB anisotropies in a non-flat Friedmann universe, relying on a very stable algorithm for the calculation of hyperspherical Bessel functions, that can be pushed to arbitrary precision levels. We also introduce a new approximation scheme which gradually takes over in the flat space limit and leads to significant reductions of the computation time.Our method is implemented in the Boltzmann code class. It can be used to benchmark the accuracy of the camb code in curved space, which is found to match expectations. For default precision settings, corresponding to 0.1% for scalar temperature spectra and 0.2% for scalar polarisation spectra, our code is two to three times faster, depending on curvature. We also simplify the temperature and polarisation source terms significantly, so the different contributions to the Cℓ 's are easy to identify inside the code.

Study of few body Kaonic Nuclei using the method of hyperspherical functions in momentum representation

Study of few body Kaonic Nuclei using the method of hyperspherical functions in momentum representation

Microscopic investigation of the structure characteristics and wave functions of the five-body hypernucleusHeλ5

The Parentage Scheme of Summarization to the N-body symmetrized basis construction (1), necessary for the description of the structural characteristics and decay reactions of the hypernuclear and nuclear systems with arbitrary amount of particles, is applied to the solution of five-body problem in hypernuclear physics. Hypernucleus �� � � as a system of four nucleons and one hyperon is investigated by the use of the Hyperspherical Function Method in momentum space. The dependence of the structure characteristics and wave functions on the types of nucleon-nucleon and hyperon-nucleon interaction potentials is studied. Mean square�� � − � � �and�� − � distances and the binding energies for the �� and ��� � � are obtained.

Investigation of the structure of the few body kaonic nuclei using the method of hyperspherical functions in momentum space

The kaonic clusters KNN and KNNN are studied within the method of hyperspherical functions in the momentum representation. Binding energies and widths of three- and four-body nuclear quasibound states are calculated by employing realistic NN potentials and the energy dependent chiral KN interaction, as well as a phenomenological KN potential. The comparison of the results of calculations obtained in the framework of the variational method, Faddeev, Faddeev-Yakubovsky equations, and method of hyperspherical functions in configuration space are discussed.

Use of generalized hyperspherical Sturmian functions for a three-body break-up model problem

A hyperspherical Sturmian approach recently developed for three-body break-up processes is tested through an analytically solvable S-wave model. The scattering process is represented by a non-homogeneous Schrödinger equation in which the driven term is given by a Coulomb-like interaction multiplied by the product of a continuum wavefunction and a bound state in the particles' coordinates. The model contains most of the difficulties encountered in a real three-body scattering problem, e.g., non-separability in the electrons' spherical coordinates and Coulombic asymptotic behaviour, and thus provides an interesting benchmark for numerical methods. Since the Sturmian basis functions are constructed so as to include the correct asymptotic behaviour, a very fast convergence of the scattering wavefunction is observed. Excellent agreement is found with the analytical results for the associated transition amplitude. This holds true down to very low energies, a domain which is usually challenging as it involves huge spatial extensions. Within our method, such calculations can be performed without increasing significantly the computational requirements.

A Simple Approach to Study the Isospin Effect in Mass Splitting of Three-Nucleon Systems by Using Hyperspherical Functions

In this work, the binding energy and wavefunctions of three-nucleon systems are obtained by using hyperspherical harmonic approach. We have used a mathematical modification method to obtain the eigenvalues and eigenfunctions of Schrödinger equation for three-nucleon systems in calculation. Next, we have used a simple approach to obtain the difference between binding energy of 3H and 3He where gives us mass splitting of three-nucleon systems. We have compared our results with the other works and experimental values.

Orthogonal and diagonal dimension fluxes of hyperspherical function

Abstract In this paper, we present the theoretical research results of certain characteristics of the generalized hyperspherical function with two degrees of freedom as independent dimensions. Here, we primarily give the answers to the quantification of dimensional potentials (fluxes) of this function in the domain of natural numbers. In addition, we also give the solutions to continual fluxes of separate contour hyperspherical (HS) functions. The symbolical evaluation and numerical verification of the values of series and integrals are realized using MathCAD Professional and Mathematica. MSC 2010: 51M04; 33E99.

On the Method of N-Body Hyperspherical Basis Symmetrization

Parentage Scheme of Summarization (PSS) to the N-body symmetrized basis construction necessary for the description of the structural characteristics and decay reactions of the hypernuclear and nuclear systems with arbitrary amount of particles is introduced. Proposed method allows to construct N-particle symmetrized hyperspherical functions on the bases of N-particle hyperspherical functions symmetrized with respect to N–1-particles by the use of the Kinematic Rotation Coefficients (KRC) related with the (N-1)-th and N-th particle permutations. The main problem that arises when investigating dynamics of few-body systems in physics is the problem of kinematic rotations under particle permutations. When number of particles increases, kinematic rotations include not only particle permutation but also transitions between different possible configurations, and mathematical calculations using complex general formula become impossible. Moreover, no general formula exists for systems with more than four particles. In order to solve kinematic rotation problem for N-particle systems, the recurrence method of determination of the KRC is applied. According to this method, the initial coefficients with the lowest quantum numbers are calculated by solving the overlap integral analytically, wave functions with arbitrary quantum numbers are expanded in terms of the basic hyperspherical functions, and the kinematic rotations of the obtained expansion are performed with the use of already known coefficients with the lowest quantum numbers. Significant advantage of the recurrence method is that no principal difficulties arise when increasing number of particles. Furthermore, recurrence relations contain numerical coefficients that are easy to evaluate by substituting appropriate quantum numbers.The problem of symmetrization of N-Body hyperspherical functions is solved by the use of the PSS. A construction scheme is given for the symmetrized hyperspherical basis in (4+1) and (5+1) configurations. The relations between parentage coefficients and the KRC of the basic hyperspherical functions are obtained. Parentage coefficients for N-Body systems are introduced. It is demonstrated that no principal difficulties arise when increasing number of particles.

Three-body radio-frequency association of Efimov trimers

We present a theoretical analysis of rf association of Efimov trimers in a 2-component Bose gas with short-range interactions. Using the adiabatic hyperspherical Green's function formalism to solve the quantum 3-body problem, we obtain universal expressions for 3-body rf association rates as a function of the s-wave scattering length $a$. We find that the association rates scale as $a^{-2}$ in the limit of large $a$, and diverge as $a^3 a_{ad}^{3}$ whenever an Efimov state crosses the atom-dimer threshold (where $a_{ad}$ stands for the atom-dimer scattering length). Our calculations show that trimer formation rates as large as $\sim10^{-21}$ cm$^6$/s can be achieved with rf Rabi frequencies of order 1 MHz, suggesting that direct rf association is a powerful tool of making and probing few-body quantum states in ultracold atomic gases.

The Dimensional Fluxes of the Hypercylindrical Function

We present the results of the theoretical researches of some characteristics of the generalized hyperspherical function with two degrees as independent dimensions. Primarily are given the answers on the quantification of the dimensional potentials (fluxes) of this function in the domain of the integer natural numbers. Beside them, we have got the solutions for some continual fluxes of the contour hypercylindrical (HY) functions on the basis. The symbolical evaluation and numerical verification of the series values and integrals are realized through the program packages Mathcad Professional and Mathematica.