Low-energy three-body scattering in a hyperspherical adiabatic basis

We propose a hyperspherical adiabatic formalism for the calculation of the 3-to-3S-matrix at low energy, for repulsive potentials, and use it then in a model calculation. That is for McGuire's model (3 particles in one dimension subject to repulsive delta-function interactions), we use analytical expressions for the hyperspherical adiabatic basis, the adiabatic coupling matrix elements, and eigenpotentials to obtain the first terms of the exactS-matrix analytically, in an expansion in powers of the wave number. We were able to associate the definite powers ofq in the expansion of theS-matrix to the corresponding inverse powers of in the expansions of the adiabatic eigenpotentials and coupling matrix elements. We investigate the effect of making the usual approximations found in the literature (extreme and uncoupled adiabatic approximations), when calculating the diagonal and off-diagonalS-matrix elements. Finally, we show that the coupled adiabatic equations uncouple as the energy goes to zero.     

Continuum three-body states using the hyperspherical adiabatic basis set

In this paper we investigate the feasibility of employing the hyperspherical adiabatic (HA) basis set to describe continuum three-nucleon states. In particular, the HA expansion is compared with the simpler expansion on hyperspherical harmonics (HH). A practical numerical application is presented using the MT-III potential.     

Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem

A hyperspherical Coulomb spheroidal (HSCS) representation is proposed for the Coulomb three-body problem. This is a new expansion in the set of well-known Coulomb spheroidal functions. The orthogonality of Coulomb spheroidal functions on a constant-hyperradius surface ρ = const rather than on a constant-internuclear-distance surface R = const, as in the traditional Born-Oppenheimer approach, is a distinguishing feature of the proposed approach. Owing to this, the HSCS representation proves to be consistent with the asymptotic conditions for the scattering problem at energies below the threshold for three-body breakup: only a finite number of radial functions do not vanish in the limit of ρ→∞, with the result that the formulation of the scattering problem becomes substantially simpler. In the proposed approach, the HSCS basis functions are considerably simpler than those in the well-known adiabatic hyperspherical representation, which is also consistent with the asymptotic conditions. Specifically, the HSCS basis functions are completely factorized. Therefore, there arise no problems associated with avoided crossings of adiabatic hyperspherical terms.     

Adiabatic hyperspherical approach to the Coulomb three-body problem: Theory and numerical method

The adiabatic hyperspherical (AH) approach to the three-body Coulomb bound-state problems is considered. The variational method of computation of the AH harmonics potential curves and coupling matrix elements is developed. The method takes into account the asymptotic behaviour of the AH harmonics at large and small values of the hyperradius . The developed method allows to perform calculations with high accuracy and stability for any hyperradius (0,) with only a few AH harmonics. The efficiency of the method and its convergence is illustrated by calculations of energy levels of the mesic moleculesdd anddt.     

On the three-body Coulomb scattering problem

We describe the smoothness properties and the asymptotic form of the Green's function (in configuration space) for three charged particles. We also discuss the integral equations and the boundary value problems for the Coulomb wavefunctions and we show that they form a complete set. Finally, we study the singularities of the Coulomb scattering operator, and we investigate the connection between the Dollard wave operators and the Coulomb wavefunctions.     

Elastic scattering, muon transfer, bound states and resonances in the three-body mesic molecular systems in the reduced adiabatic hyperspherical approach

The uniform method of numerical investigation of bound states and scattering processes 2→ 2 (including resonance states) in the Coulomb three-body (CTB) systems is developed. It is based on the adiabatic hyperspherical approach (AHSA) and includes the numerical realization and applications to the three-body mesic atomic systems. The results of calculations of bound states of these systems (including the local characteristics of the wave functions) and the scattering processes 2→ 2 (including the characteristics of the resonance states) are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

A solution of the Coulomb three-body problem

In this work, a final state wave function is constructed which represents a solution of the three-body Schr?dinger equation. The formulated wave function is superimposed of one basic analytical function with various parameters. The coefficients of these basic functions involved in final state wave function can be easily calculated from a set of linear equations. The coefficients depend only on incident energy of the system. The process can also be prolonged for application to the problems more than three bodies.     

Matrix elements of two-body operators between three-body hyperspherical basis states

Matrix elements of two-body operators between three-body spherically symmetric hyperspherical basis states are expressed through the Clebsch-Gordon coefficients and explicitly written functions of a radial variable.     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e ^?, e ^?, He⁺⁺) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.     

Solution of three-dimensional Faddeev equations for three-body Coulomb bound states

A new method of directly solving the three-dimensional Faddeev equations in the total-angular-momentum representation for the pure Coulomb bound-state problem is developed. The method is based on the tri-quintic Hermite spline expansion of the Faddeev components. The ground states of thee ^– e ^– e ⁺ system and thepp ^– mesic molecule are calculated.     

New effective adiabatic approach to the muonic three-body scattering problem

The effective adiabatic approach to analysis of the three-particle interaction is presented. It gives a possibility to represent even in a simple two-level approximation all qualitative peculiarities of mesic atomic resonance reactions and to obtain a good quantitative agreement with different cumbersome calculations.     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

Classical dynamics near the triple collision in a three-body Coulomb problem

We investigate the classical motion of three charged particles with both attractive and repulsive interactions. The triple collision is a main source of chaos in such three-body Coulomb problems. By employing the McGehee scaling technique, we analyze here for the first time in detail the three-body dynamics near the triple collision in 3 degrees of freedom. We reveal surprisingly simple dynamical patterns in large parts of the chaotic phase space. The underlying degree of order in the form of approximate Markov partitions may help in understanding the global structures observed in quantum spectra of two-electron atoms.     

Application of the hyperspherical formalism to the trinucleon bound state problems

A method for solving the Schrödinger equation for the ground state of any number of bosons or for the trinucleon system or α-particle is formulated in the framework of the hyperspherical harmonic expansion method. It is applied to the trinucleon system for nucleons interacting through realistic soft core potentials. The convergence of the method is carefully studied. Binding energies, electric form factors, one body densities, two body correlation functions and two body photodisintegration are calculated for various potentials.     

Coulomb forces in the three-body problem with application to direct nuclear reactions

Faddeev equations are considered in the case of three charged particles interacting with both separable nuclear two-body interactions and also including Coulomb forces. Modified Faddeev equations with Coulomb Green's functions are introduced. The three-body amplitudes are given into pure Coulomb and distorted-Coulomb amplitudes. Introducing a decomposition in the angular momentum states, a set of three-body integral equations is obtained. The effect of pure coulomb amplitudes is studied in direct nuclear reactions and found to give a large contribution to the cross sections. The three-body integral equations obtained are applied for direct nuclear reactions. The angular distributions for¹²C(⁶Li,d)¹⁶O,¹⁶O(⁶Li,d)²⁰Ne, and¹²C(⁶Li,α)¹⁴N transfer reactions are calculated as well as for the⁶Li elastic scattering on¹²C. From the good agreement between the theoretically calculated and experimental data, better spectroscopic factors are extracted. The effect of including Coulomb forces in the three-body problem is found to improve the results by about 16.26%.     

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

We present a non-variational approach to the solution of the quantum three-body problem, based on the decomposition of the three-body Laplacian operator through the use of its intrinsic symmetries. With the judicious choice of angular momentum eigenfunctions, a clean separation of spatial rotation from kinematic rotation is achieved, leading to a finite set of coupled PDEs in terms of the canonical variables. Numerical implementation of this approach to the three-body Coulomb problem is shown to yield accurate ground state eigenvalues and wavefunctions, together with those of low-lying excited states. We present results on some typical three-body systems. In particular, the eigenvalues and wavefunctions of the even-parity state of the negative hydrogen ion are detailed for the first time. The issue of computational efficiency is also briefly discussed.     

Some properties of bound states in the three-body problem with separable potentials

A system of three identical bosons with the pair separable S-interaction is considered. The monotonic (Yamaguchi potential) and oscillating pair potentials for the three-alpha model of¹²C have been investigated. In the case of the Yamaguchi potential many excited states (second, third, etc.) have been found.The author is thankful to Prof. V. G. Neudatchin and Prof. Yu. F. Smirnov for valuable discussions on the problems concerned.