Beyond the first order of the Hyperspherical Harmonic Expansion Method

In this paper we demonstrate the inadequacy of the first order of the Hyperspherical Harmonic Expansion Method, the L_m approximation, for the calculation of the binding energies, charge form factors and charge densities of doubly magic nuclei like ¹⁶O and ⁴⁰Ca. We then extend the Hyperspherical Expansion Method to many-fermion systems, consisting of an arbitrary number of fermions, and develop an exact formalism capable of generating the complete optimal subset of the hyperspherical harmonic basis functions. This optimal subset consists of those hyperspherical harmonic basis functions directly connected to the dominant first term in the expansion, the hyperspherical harmonic of minimal order L_m, through the total interaction between the particles. The required many-body coefficients are given using either the Gogny or Talmi-Moshinsky coefficients for the two-body operators. Using the two-body coefficients the weight function generating the orthogonal polynomials associated with the optimal subset is constructed.     

Structure and Occurrence of Three-Body Halos in Two Dimensions

 The quantum-mechanical three-body problem is reformulated in two dimensions by use of hyperspherical coordinates and an adiabatic expansion of the Faddeev equations. The effective radial potentials are calculated and their large-distance asymptotic behavior is derived analytically for short-range two-body interactions. Energies and wave functions are computed numerically for various potentials. An infinite series of Efimov states does not exist in two dimensions. Borromean systems, i.e. bound three-body systems without bound binary subsystems, can only appear when a short-range repulsive barrier at finite distance is present in the two-body interaction. The corresponding Borromean state is never spatially extended. For a system of three weakly interacting identical bosons we find two bound states with both binding energies proportional to the two-body binding energy. In the limit of small binding these states are spatially located at the very large distances characterized by the scattering length. Their properties are universal and independent of the details of the potential. We compare throughout with the corresponding properties in three dimensions. Received September 25, 1998; accepted for publication January 30, 1999     

Recurrence relations between transformation coefficients of hyperspherical harmonics and their application to moshinsky coefficients

Closed formulae and recurrence relations for the transformation of a two-body harmonic oscillator wave function to the hyperspherical formalism are given. With them Moshinsky or Smirnov coefficients are obtained from the transformation coefficients of hyperspherical harmonics. For these coefficients the diagonalization method of Talman and Landéreduces to simple recurrence relations which can be used directly to compute them. New closed formulae for these coefficients are also derived : they are needed to compute the two simplest coefficients which determine the sign for the recurrence relation.     

Ground state baryons in the non-relativistic quark model

We analyse the properties of the ground state bayons in the non-relativistic quark model. The three-body problem is solved by means of the hyperspherical expansion. We consider various two-body potentials of power law type and also a three-body linear potential. In displaying the results, we insist on quantities like ratios of splittings which are scale independent and are functions only of the power of the potential and of the ratio of quark masses.     

The first order of the hyperspherical harmonic expansion method

The hyperspherical harmonic expansion method is studied in this work. Our attention is focused on the properties of the L_m-approximation in which only the hyperspherical harmonics of minimal order are taken into account. Exact solutions of the Schrödinger equation for a few simple hyperspherical potentials are given. Recipes for constructing antisymmetric hyperspherical harmonics for fermions are investigated, and various procedures to derive the effective potential in the L_m-approximation are discussed. The method is applied to the calculation of ground state and hyperradial excited states (which are identified as the breathing modes) of doubly-magic nuclei. Finally, the energy per particle is derived in the L_m-approximation with Skyrme like forces for an infinitely heavy self-conjugate nucleus.     

Electromagnetic and weak transitions in light nuclei

Recent advances in the study of the p-d radiative and He weak capture processes by our group are presented and discussed. The trinucleon bound and scattering states have been obtained from variational calculations by expanding the corresponding wave functions in terms of correlated hyperspherical harmonic functions. The electromagnetic and weak transition currents include one- and two-body operators. The accuracy achieved in these calculations allows for interesting comparisons with experimental data.Received: 1 November 2002, Published online: 15 July 2003PACS: 21.45.+v Few-body systems - 23.40.-s Beta decay; double beta decay; electron and muon capture - 25.40.Lw Radiative capture     

Calculation of the α-Particle Ground State

The correlated hyperspherical harmonic expansion method is used to calculate α-particle properties with a realistic Hamiltonian consisting of the Argonne V14 two-nucleon and Urbana model VIII three-nucleon potentials. The calculated binding energy, mass radius and wave percentages are close to the corresponding quantities obtained with Green's-function Monte-Carlo and Faddeev-Yakubovsky techniques. Received June 29, 1994; accepted for publication August 12, 1994     

An integro-differential equation approach to the many-body problem

An integro-differential equation in two variables which includes two-body correlations, has been derived for a many-body system of identical particles. This equation has been obtained by means of a complete expansion in terms of the hyperspherical harmonic potential basis. It is proved that this equation is identical to the Faddeev equation for a system of three bosons. The implications of this result for many-body systems are discussed.     

Correlated hyperspherical-harmonic expansion for three-nucleon systems

The hyperspherical-harmonic-expansion method is applied to solve the Schrödinger equation for a three-particle system interacting via central spin-dependent potentials. The convergence of the expansion has been improved by multiplying the hyperspherical basis by an appropriate correlation factor, chosen as a product of one-dimensional functions fixed by means of a two-body Schrödinger equation. The results obtained for three nucleons interacting via the Malfliet-Tjon potential are in close agreement with those given by the most accurate methods.     

The potential harmonic expansion method

Various properties of the hyperspherical potential basis are investigated. The expansion of any two-body function, in particular the two-body potential, is given. The matrix elements with two and three potential harmonics needed for the construction of the potential matrix are calculated. Useful recurrence formulae are derived. The concept of potential basis is extended to systems with any number of fermions. A method for improving the accuracy of the expansion of the wavefunction by taking into account more than the two-body correlations is suggested.     

Condensates and Correlated Boson Systems

We study two-body correlations in a many-boson system with a hyperspherical approach, where we can use arbitrary scattering length and include two-body bound states. As a special application we look on Bose-Einstein condensation and calculate the stability criterium in a comparison with the experimental criterium and the theoretical criterium from the Gross-Pitaevskii equation.     

Relativistically corrected masses of ground-state baryons in hyperspherical harmonic quark model

The masses of the ground-state light baryons are calculated in the quark model. The unperturbed wave functions correspond to a hyperspherical harmonic confinement. The perturbation includes a short-range potential and all the relevant relativistic corrections of the order ofv ²/c ². Results are compared with the experimental values and found to be in good agreement. This may be a test not of the hyperspherical harmonic model so much as of the feasibility of a simple (but consistent) relativistically corrected fit of the light baryon masses.     

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.     

Method for solving the many-body bound state nuclear problem

We present a method based on hyperspherical harmonics to solve the nuclear many-body problem. It is an extension of accurate methods used for studying few-body systems to many bodies and is based on the assumption that nucleons in nuclei interact mainly via pairwise forces. This leads to a two-variable integro-differential equation which is easy to solve. Unlike methods that utilize effective interactions, the present one employs directly nucleon-nucleon potentials and therefore nuclear correlations are included in an unambiguous way. Three body forces can also be included in the formalism. Details on how to obtain the various ingredients entering into the equation for the A-body system are given. Employing our formalism we calculated the binding energies for closed and open shell nuclei with central forces where the bound states are defined by a single hyperspherical harmonic. The results found are in agreement with those obtained by other methods.     

Effect of the three-body force on trinucleon bound systems consideringS andS′-state admixture

We report the calculation of binding energy, charge form factor and point-like proton density of both³H and³He by the hyperspherical harmonics method with the inclusion of two-pion exchange three-nucleon force (Fujita-Miyazawa type). For the two-body force theN-N Afnan-Tang S-3 potential is taken. Coulomb and three-body forces are treated nonperturbatively. In this calculation the mixed symmetryS′-state of the trinucleon ground state is considered along with the space totally symmetricS-state.     

A set of hyperspherical harmonics especially suited for three-body collisions in a plane

We obtain a set of four-dimensional hyperspherical harmonics in closed form. These harmonics are not only quantized with respect to the rotation group (O ₂), but are an irreducible basis for the permutation groupS ₃. An additional symmetry is found which allows us to write hyperspherical harmonics classified with respect to a 12 element groupS ₃×i×O ₂. We give a set of three mutually commuting operators whose eigenvalues uniquely characterize each spherical harmonic with respect to degree, symmetry, and angular momentum in the plane.     

Low-Lying S-States of Two-Electron Systems

The energies of the low-lying bound S-states of some two-electron systems (treating them as three-body systems) like negatively charged hydrogen, neutral helium, positively charged-lithium, beryllium, carbon, oxygen, neon, argon and negatively charged muonium and exotic positronium ions have been calculated employing hyperspherical harmonics expansion method. The matrix elements of two-body interactions involve Raynal–Revai coefficients which are particularly essential for the numerical solution of three-body Schr?dinger equation when the two-body potentials are other from Coulomb or harmonic. The technique has been applied for to two-electron ions ¹H^? (Z = 1) to ⁴⁰Ar¹⁶⁺ (Z = 18), negatively charged-muonium Mu^? and exotic positronium ion Ps^?(e ⁺ e ^? e ^?) considering purely Coulomb interaction. The available computer facility restricted reliable calculations up to 28 partial waves (i.e. K _m  = 28) and energies for higher K _m have been obtained by applying an extrapolation scheme suggested by Schneider.     

类Quesne环状球谐振子势场中的赝自旋对称性

提出了一种新的类Quesne环状球谐振子势,应用二分量方法求解1/2-自旋粒子满足的Dirac方程, Dirac哈密顿量由标量和矢量类Quesne环状球谐振子势构成.在Σ=S(r)+V(r)=0的条件下,得到了Dirac旋量波函数下分量的束缚态解和能谱方程, 显示出类Quesne环状球谐振子势场中的赝自旋对称性.讨论了束缚态波函数和能谱方程的有关性质. 关键词： 类Quesne环状球谐振子势 Dirac方程 赝自旋对称性 束缚态     

Calculating the characteristics of neutron-deuteron and proton-deuteron systems in a two-body potential model

³H and ³He nuclei are considered in a two-body model (³H = n + d; ³He = p + d). Two independent approaches are used: in the first, interaction is described by the folding potential, while NN potentials are taken in the Hulthen form with allowance for violations of isotopic invariance. The second approach features phenomenological Hulthen and Yukawa Nd potentials used as Nd interaction. In both approaches, the binding energies, vertex constants, and asymptotic normalization coefficients in the N+d channel are calculated.