Numerical Methods in Configuration-Space A = 3, 4 Bound-State and Scattering Calculations

Schellingerhout, N. W.

doi:10.1007/978-3-7091-9352-5_50

Few-Body Problems in Physics ’93

1994

DOI: 10.1007/978-3-7091-9352-5_50

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Numerical Methods in Configuration-Space A = 3, 4 Bound-State and Scattering Calculations

N. W. Schellingerhout

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1998

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“…For comparison we give the corresponding result of Faddeev calculations of Ref. [2,13], where channel-by-channel comparison is possibble, and of Ref. [4], where this kind of comparison is not possibble, and also of results of a very accurate variational calculation [14].…”

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confidence: 99%

Three-Potential Formalism for the Atomic Three-Body Problem

Papp

1998

Few-Body Systems

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Based on a three-potential formalism we propose mathematically wellbehaved Faddeev-type integral equations for the atomic three-body problem and descibe their solutions in Coulomb-Sturmian space representation. Although the system contains only long-range Coulomb interactions these equations allow us to reach solution by approximating only some auxiliary short-range type potentials. We outline the method for bound states and demonstrate its power in benchmark calculations. We can report a fast convergence in angular momentum channels.The Faddeev equations are the fundamental equations of the three-body problems. Besides giving a unified formulation, they are superior to the Schrödinger equation in many respects: in incorporating the boundary conditions, in treating the symmetries, in handling the correlations, etc.. Nevertheless, their use in atomic three-body calculations is rather scarce [1][2][3][4], and these calculations showed that to reach a reasonable accuracy many channels are needed. So, the belief spread in the community that the Faddeev equations, the fundamental equations of three-body systems, are not well-suited for treating atomic threebody problems and other techniques can do a much better job, at least for bound states. In Refs. [1-4] the Faddeev equations were used in such a form that, the solution could be reached only by some kind of approximation on the long-range Coulomb potential, and thus the convergence were ensured only via the square integrability of the bound-state wave function.The aim of this paper is to solve the atomic three-body problems by approximating only short-range type interactions. We invoke a newly established "three-potential" formalism and derive such kind of Faddeev-type integral equations which contain only short-range type interactions as source terms. We solve the equations by approximating only the short-range type source terms. We point out that although we are working with finite matrices the wave functions possesses correct three-body Coulomb asymptotics. Finally, as compulsory benchmark cases, we calculate the helium atom, the positronium ion and the muonic hydrogen molecule ion, and will observe a fast convergence with respect to angular momentum channels.The "three-potential" formalism was designed for solving nuclear three-body problems in the presence of Coulomb interaction. The method was presented first in bound-sate calculations [5] and was extended to below-breakup scattering calculations [6] where the 1

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Three-Potential Formalism for the Atomic Three-Body Problem

Papp

1998

Few-Body Systems

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Four-body bound states in momentum space: the Yakubovsky approach without two-body t − matrices

Mohammadzadeh,

Radin,

Mohseni

et al. 2023

Front. Phys.

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This study presents a solution to the Yakubovsky equations for four-body bound states in momentum space, bypassing the common use of two-body t − matrices. Typically, such solutions are dependent on the fully-off-shell two-body t − matrices, which are obtained from the Lippmann-Schwinger integral equation for two-body subsystem energies controlled by the second and third Jacobi momenta. Instead, we use a version of the Yakubovsky equations that does not require t − matrices, facilitating the direct use of two-body interactions. This approach streamlines the programming and reduces computational time. Numerically, we found that this direct approach to the Yakubovsky equations, using 2B interactions, produces four-body binding energy results consistent with those obtained from the conventional t − matrix dependent Yakubovsky equations, for both separable (Yamaguchi and Gaussian) and non-separable (Malfliet-Tjon) interactions.

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Метод Пакетной Дискретизации Континуума Для Решения Трехчастичной Задачи Рассеяния

Kukulin¹,

Pomerantsev²,

Rubtsova³

2007

ТМФ

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Numerical Methods in Configuration-Space A = 3, 4 Bound-State and Scattering Calculations

Cited by 3 publications

References 5 publications

Three-Potential Formalism for the Atomic Three-Body Problem

Three-Potential Formalism for the Atomic Three-Body Problem

Four-body bound states in momentum space: the Yakubovsky approach without two-body t − matrices

Метод Пакетной Дискретизации Континуума Для Решения Трехчастичной Задачи Рассеяния

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