Sensitivity of the three-body calculations to different effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system³H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.     

Study of various few-body systems using Gaussian expansion method (GEM)

We review our calculation method, Gaussian expansion method (GEM), to solve accurately the Schrödinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body (3- to 5-body) systems, such as 1) few-nucleon systems, 2) few-body structure of hypernuclei, 3) clustering structure of light nuclei and unstable nuclei, 4) exotic atoms/molecules, 5) cold atoms, 6) nuclear astrophysics and 7) structure of exotic hadrons. Showing examples in our published papers, we explain i) high accuracy of GEM calculations and its reason, ii) wide applicability of GEM to various few-body systems, iii) successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for shortrange and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes (oscillations). These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.     

Dependence of the Three-Nucleon Binding Energy on Some Different Effects

The bound state of three-nucleon system is studied as a three-body problem which is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. The three-body problem is reduced to a set of coupled integral equations by using separable approximations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system ³H. The main interest of the present work is to investigate the sensitivity of the three-nucleon binding energy to different effects. For this reason, we study the dependence of this energy on different forms of local and separable nucleon-nucleon potentials, the effective range of the nucleon-nucleon interaction, and on the percent of the D state in the deuteron wave function. Also we test the sensitivity of the three-nucleon binding energy to the considered number of terms from the separable expansion.     

Quality of the three-nucleon bound-state wave function from a two-nucleon separable expansion method

The numerical quality of the³H wave function obtained by the separable expansion method of Ernst, Shakin, and Thaler is examined. Separable approximations to the Paris potential with increasing accuracy are used in the¹ S ₀ and³ S ₁-³ D ₁ partial waves to calculate the binding energy, wave function, wave-function component percentages, and theS- andD-wave asymptotic normalization constants of³H. The results are compared with existing five-channel calculations obtained directly (without expansion) from the Paris potential to determine convergence. It is found that the results converge rapidly to the right values, indicating that the³H wave function thus obtained is of high quality and essentially indistinguishable from that obtained directly from the Paris interaction.Dedicated to Profs. Erich Schmid and Ivo laus on the occasion of their 60th birthdays     

Comparison of Triton Bound-State Properties Using Different Separable Representations of Realistic Nucleon-Nucleon Potentials

 The quality of two different separable expansion methods (W-matrix and Ernst-Shakin-Thaler) is investigated. We compare the triton binding energies and components of the triton wave functions obtained in this way with the results of a direct two-dimensional treatment. The Paris, Bonn A and Bonn B potentials are employed as underlying two-body interactions, their total angular momenta being incorporated up to . It is found that the most accurate results based on the Ernst-Shakin-Thaler method agree within 1.5% or better with the two-dimensional calculations, whereas the results for the W-matrix representation are less accurate. Received February 9, 1999; accepted for publication February 13, 1999     

Perturbation theory for QED bound states

A new method is presented for deriving a systematic perturbative expansion for QED bound states, which does not rely upon solving any new or old equation. The starting point is a given nonperturbative zeroth order Green's function, obtained by a suitable “relativistic dressing” of the nonrelativistic Green's function for the Schrödinger equation with Coulomb potential, which embodies the Coulombic bound states and is known. The comparison with the complete Green's function as given by standard perturbative QED gives a perturbative kernel which is then used for the expansion of the QED Green's function in terms of the given non-perturbative zeroth order Green's function.     

On the validity and properties of the atom-atom potential as a function of intermolecular separation,configuration, and partial wave order

The intermolecular partial wave expansion of the atom-atom potential U is reviewed briefly and developed, by using results due to Sack, so that the radial components of the expansion can be evaluated to arbitrary accuracy for all relevant partial wave orders and values of the intermolecular distance r. These results are used to study the convergence of the partial wave expansion of U as a function of partial wave order, r, intermolecular orientation, and the anisotropy of the interacting molecules. In marked contrast to previous work it is found that many of the higher order partial wave components of U are important relative to the isotropic term even for the interaction of relatively spherical molecules and that the results obtained from a truncated partial wave expansion depend significantly upon the method of summation due to the generally poor convergence of the expansion. The validity of the atom-atom potential as a representation of the correct attractive intermolecular potential is also discussed in some detail. There are basic problems associated with the representations furnished by both the isotropic and the anisotropic parts of the atom-atom potential at intermediate and large r. The different convergence properties of the r ^-1 expansions of the partial wave expansions of U and of the correct potential for these values of r is illustrated by using model interactions. While it appears that it may be possible to obtain a qualitatively reasonable representation of the attractive part of an intermolecular potential over a useful range of r from atom-atom results, this apparently cannot be achieved for wider ranges of r or for the purely anisotropic part of the potential.     

Viable improvement of the full potential linearized augmented plane wave (FLAPW) method

A characteristic feature of the Full Potential Linearized Augmented Plane Wave (FLAPW)-method consists in the spatial subdivision of the charge density analogous to that of the one-particle wavefunctions, i.e. into a portion that is expanded in terms of spherical harmonics Y_lm inside the muffin-tin spheres and into a plane wave expansion of the interstitial charge density. To obtain the Hartree potential inside the spheres one is hence forced to solve a boundary value problem at the sphere surface. In addition, in all non-equivalent spheres each (l,m)-component of the charge density is mapped onto 300-400 radial grid points. To ensure an accelerated convergence of the calculation, the pertinent schemes require this rather large data set to be stored and mixed within 3-6 iteration steps. We show and illustrate for the example of a spin-polarized Ni-film with and without an oxygen overlayer and for bulk Si that this data set can be compressed by at least two orders of magnitude if one partitions the charge density in a different way so that the relevant portion determining the interatomic bonding can be Fourier expanded throughout the lattice cell. One thereby arrives at a modified FLAPW-scheme that combines favorable features of the original method with virtues of the pseudopotential method which consist in the simple construction of the Hartree potential and the efficient way of achieving self-consistency. These advantages can be exploited to the fullest by using Fast Fourier Transform. Moreover, forces that atoms experience in off-equilibrium positions attain a particularly simple form in terms of the charge density expansion coefficients. Received: 31 May 1997 / Revised: 12 December 1997 / Accepted: 18 December 1997     

Relativistic addition of interactions in constraint Hamiltonian mechanics

The problem of constructing separable N-particle interactions for given two-body forces in the constraint Hamiltonian approach to relativistic classical mechanics is tackled in two different ways. A formal expression for the N-particle constraints is given in terms of a classical wave operator. An (independent) iterative scheme for these constraints is carried out. Explicit formulas for the three-particle interactions are written down, accurate up to second order in the two-body coupling constants and symmetric with respect to particle permutations.     

Global-Vector Representation of the Angular Motion of Few-Particle Systems

The angular motion of a few-body system is described by introducing angles that depend on the positions of the particles. The most adequate angles are determined variationally. The numerical examples presented for the molecule, Li atom, antiprotonic helium atom, triton and alpha particle convincingly show that this method is an excellent alternative of the partial-wave decomposition of the wave function. Received November 4, 1997; accepted in final form January 19, 1998     

A generalized Talmi-Moshinsky transformation for few-body and direct interaction matrix elements

A set of basis states for use in evaluating matrix elements of few-body system operators is suggested. These basis states are products of harmonic oscillator wave functions having as arguments a set of Jacobi coordinates for the system. We show that these harmonic oscillator functions can be chosen in a manner that allows such a product to be expanded as a finite sum of the corresponding products for any other set of Jacobi coordinates. This result is a generalization of the Talmi-Moshinsky transformation for two equal-mass particles to a system of any number of particles of arbitrary masses. With the help of our method the multidimensional integral which must be performed to evaluate a few-body matrix element can be transformed into a sum of products of three dimensional integrals. The coefficients in such an expansion are generalized Talmi-Moshinsky coefficients. The method is tested by calculation of a matrix element for knockout scattering for a simple three-body system. The results indicate that the method is a viable calculational tool.     

A separable parametrization of the trinucleon wave function with the Reid soft-core potential

We present a useful analytical parametrization of the bound trinucleon wave function obtained from solving the Faddeev equations with the Reid soft-core potential. The partial wave components of the wave function are expanded in separable terms having rational fraction forms which incorporate the appropriate asymptotic behaviors. Both momentum and configuration space representations of the parametrized partial wave components are presented.     

强光场下电子与库仑势散射及多光子过程

对Schr?dinger方程引入修饰势,并将波函数展开为Floquet分波,即可实施分离变量。强光场时弱耦合法选取合适微扰量,径向波动方程可逐步近似求解。以圆极化强光场为例,计算了散射波函数、散射矩阵和截面。重要结论是:强光场下将出现多光子过程,对应m次多光子过程有相应的共振谱线族,谱线能量约为m?ω,并给出谱线强度计算公式。 关键词：     

Eikonal analysis of Coulomb distortion in quasi-elastic electron scattering

An eikonal expansion is used to provide systematic corrections to the eikonal approximation through order 1/k ², where k is the wave number. Electron wave functions are obtained for the Dirac equation with a Coulomb potential. They are used to investigate distorted-wave matrix elements for quasi-elastic electron scattering from a nucleus. A form of effective-momentum approximation is obtained using trajectory-dependent eikonal phases and focusing factors. Fixing the Coulomb distortion effects at the center of the nucleus, the often-used ema approximation is recovered. Comparisons of these approximations are made with full calculations using the electron eikonal wave functions. The ema results are found to agree well with the full calculations.     

Conformal invariance and pion wave functions of nonleading twist

The restrictions are studied for the general structure of pion wave functions of twist 3 and twist 4 imposed by the conformal symmetry and the equations of motion. A systematic expansion of wave functions in the conformal spin is built and the first order corrections to asymptotic formulae are calculated by the QCD sum rule method. In particular, we have found a multiplicatively renormalizable contribution into the two-particle wave function of twist 4 which cannot be expanded in a finite set of Gegenbauer polynonials.     

Theoretical studies of inelastic atomic collisions in a two-state model problem

A two-state scattering problem in which two non-crossing Morse curves are coupled by an exponential potential is discussed theoretically in both the diabatic and adiabatic approximations. Inelastic cross sections are estimated by various approximate analytical formulas, which are expressed in terms of the distorted wave matrix elements. The Landau (steepest descent) method is applied to estimate the distorted wave matrix element. The previously proposed separable potential approximation in the adiabatic method is found to be best among others by comparing with the exact numerical results. Transition probability as a function of l (angular momentum of relative motion) is well reproduced by this approximation. A glory effect in the velocity dependence of the cross section is found in the low energy region, and the adiabatic approximation reproduces this undulation phenomenon well.     

A variational theory of fermion matter with spin dependent correlations

A variational wave function is constructed for a system of fermions interacting with a spin dependent potential. The correlations due to the repulsive core of the potential are described by a spin independent Jastrow product ansatz and the correlations due to the longer range part of the potential, which are assumed to be spin dependent, are described by an independent pair ansatz. A cluster expansion is derived for the variational energy and a set of hypernetted chain (HNC) equations obtained to sum the cluster series in terms of the elementary diagrams. Neglecting the elementary diagrams, the HNC equations are solved numerically for the spin dependent potential, V3, in neutron matter. The introduction of spin dependent correlations is found to give a small lowering of the variational energy in the HNC approximation. The results are very sensitive to an accurate treatment of the many-body terms within the HNC approximation, however, and it is shown that additional approximations can easily lead to an exaggeration of the effect of the spin dependent correlations.     

On the validity of dwba for deuteron stripping: (I). The Mitra three-body model

Previous work showing that there exists an exact formulation of the DWBA for stripping in the S-wave, separable potential, three-body model of Mitra is discussed and extended. The one-body equation obeyed by the c.m. wave function used in the reformulated DWBA is derived and compared with the equation obeyed by the wave function used in the standard formulation of DWBA, viz., the deuteron elastic scattering wave function. Results obtained by other workers on application of three-body methods to direct reactions are discussed in light of the fact that an exact DWBA exists for the separable potential model.