A. Deltuva
The method of screening and renormalization for including the Coulomb interaction in the framework of momentum-space integral equations is applied to the three- and four-body nuclear reactions. The Coulomb effect on the observables and the ability of the present nuclear potential models to describe the experimental data is discussed.
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The $α$-proton Coulomb interaction is included in the description of $α$-deuteron scattering using the screening and renormalization approach in the framework of momentum-space three-particle equations. The technical reliability of the method is demonstrated. Large Coulomb-force effects are found.
We demonstrate the feasibility and efficiency of the Coulomb-Sturmian separable expansion method for generating accurate solutions of the Faddeev equations. Results obtained with this method are reported for several benchmark cases of bosonic and fermionic three-body systems. Correct bound-state results in agreement with the ones established in the literature are achieved for short-range interactions. We outline the formalism for the treatment of three-body Coulomb systems and present a bound-state calculation for a three-boson system interacting via Coulomb plus short-range forces. The corres…
We present a practical method to solve the proton-deuteron scattering problem at energies above the three-body breakup threshold, in which we treat three-body integral equations in coordinate space accommodating long-range proton-proton Coulomb interactions. The method is examined for phase shift parameters, and then applied to calculations of differential cross sections in elastic and breakup reactions, analyzing powers, etc. with a realistic nucleon-nucleon force and three-nucleon forces. Effects of the Coulomb force and the three-nucleon forces on these observables are discussed in comparin…
We investigate projectile breakup effects on 6Li+209Bi elastic scattering near the Coulomb barrier with the four-body version of the continuum-discretized coupled-channel method (four-body CDCC). This is the first application of four-body CDCC to 6Li elastic scattering. The elastic scattering is well described by the p+n+4He+209Bi four-body model. We propose a reasonable three-body model for describing the four-body scattering, clarifying four-body dynamics of the elastic scattering.
The method of screening and renormalization is used to include the Coulomb interaction between the charged particles in the momentum-space description of three- and four-body nuclear reactions. The necessity for the renormalization of the scattering amplitudes and the reliability of the method is demonstrated. The Coulomb effect on observables is discussed.
We propose a three-potential formalism for the three-body Coulomb scattering problem. The corresponding integral equations are mathematically well-behaved and can succesfully be solved by the Coulomb-Sturmian separable expansion method. The results show perfect agreements with existing low-energy $n-d$ and $p-d$ scattering calculations.
We present a separable expansion approximation method for Coulomb-like potentials which is based on Schwinger variational principle and uses Coulomb-Sturmian functions as basis states. The new scheme provides faster convergence with respect to our formerly used non-variational approach.
Coulomb breakup of a projectile in the Coulomb field of a fully stripped heavy nucleus is at present one of the most popular experimental methods to obtain information on reactions of interest in nuclear astrophysics. Its theoretical interpretation presents, however, considerable difficulties, due to the three-body nature and the infinite range of the Coulomb forces involved. Among the uncertainties affecting present analyses, the possible modification of the dissociation cross section by three-body Coulomb final-state interactions plays a major role. Various methods which have been proposed t…
We demonstrate numerically that proton-proton (pp) scattering observables can be determined directly by standard short range methods using a screened pp Coulomb force without renormalization. In examples the appropriate screening radii are given. We also numerically investigate solutions of the 3-dimensional Lippmann-Schwinger (LS) equation for a screened Coulomb potential alone in the limit of large screening radii and confirm analytically predicted properties for off-shell, half-shell and on-shell Coulomb t-matrices. These 3-dimensional solutions will form a basis for a novel approach to inc…
The method of screening and renormalization is used to include the Coulomb interaction between the charged particles in the description of few-body nuclear reactions. Calculations are done in the framework of Faddeev-type equations in momentum-space. The reliability of the method is demonstrated. The Coulomb effect on observables is discussed.
We propose a novel method for calculating resonances in three-body Coulombic systems. The method is based on the solution of the set of Faddeev and Lippmann-Schwinger integral equations, which are designed for solving the three-body Coulomb problem. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order that we get resonances. As a numerical illustration a toy model for the three-$α$ system is solved.
Certain comments on the paper of Yu.V. Orlov, B. F. Irgaziev and L. I. Nikitina published in Phys.At.Nuclei, {\bf 73} (2010) 757 are made.
The Coulomb Sum and Proton-Proton Correlations in Few-Body Nuclei
在核电荷算符的简单模型中，库仑和核的电荷形状因子的测量直接决定质子-质子关联。我们检查实验结果获得的几体核在贝茨和Saclay使用模型的电荷运营商，包括一个和两个身体条款。以前的分析只使用单体项，未能重现实验结果。然而，我们发现，已被用来成功地描述电荷形状因子相同的运营商也产生大量的协议与库仑和的测量。
We investigate dissociation reactions of loosely bound and spatially extended three-body systems. We formulate a practical method for simultaneous treatment of long-range Coulomb and short-range nuclear interactions. We use $^6$He (n+n+$α$) and $^{11}$Li (n+n+$^{9}$Li) as examples and study the two-neutron separation cross sections as functions of target and beam energy. Individual Coulomb and nuclear as well as interference contributions are also extracted.
Triangle Diagram with Off-Shell Coulomb T-Matrix for (In-)Elastic Atomic and Nuclear Three-Body Processes
在三体理论中，一个带电粒子离开另外两个带电粒子的束缚态的弹性和非弹性散射的驱动项包含完全离壳的两体库仑T矩阵，描述了射弹与每个带电目标粒子的中间态库仑散射。到目前为止，后者通常被替换为库仑势，无论是在使用多重散射方法或在解决三体积分方程。的精确和近似的壳驱动项的一般性质进行了讨论，这种近似的准确性进行了研究数值，无论是原子和核过程，包括束缚态激发，能量低于和高于相应的三体解离阈值，在整个范围内的散射角。
We introduce an approach, based on the coordinate space Faddeev equations, to solve the quantum mechanical three-body Coulomb problem in the continuum. We apply the approach to compute measured properties of the first two $0^+$ levels in $^{12}$C, including the width of the excited level, in a 3$α$-particle model. We use the two-body interaction which reproduces the low energy $α$-$α$ scattering data and add a three-body force to account for the compound state admixture.
We extend our approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) Faddeev calculations from elastic proton-deuteron (pd) scattering to the breakup process. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger equation. We demonstrate numerically that the proton-deuteron breakup observables can be determined from the resulting on-shell 3N amplitudes increasing the screening radius. However, contrary to the pd elastic scattering, the screening limit exists only after r…
We extend our approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) momentum-space Faddeev calculations of elastic proton-deuteron (pd) scattering and breakup to the case when also a three-nucleon force (3NF) is acting. In addition we formulate that approach in the application to electron- and gamma-induced reactions on 3He. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of a 3-dimensional Lippmann-Schwinger equation (LSE). The resulting equations have the same structure as the Faddeev equations which …
The Fock representation of the Coulomb interaction. I. Two-body problem
本文提出库仑相互作用的Fock表示，在动量空间精确可分离，无需屏蔽或拟合，并与屏蔽-重整化方法在二体基准上比较，验证其有效性。
Coulomb breakup strengths of 11Li into a three-body 9Li+n+n system are studied in the complex scaling method. We decompose the transition strengths into the contributions from three-body resonances, two-body ``10Li+n'' and three-body ``9Li+n+n'' continuum states. In the calculated results, we cannot find the dipole resonances with a sharp decay width in 11Li. There is a low energy enhancement in the breakup strength, which is produced by both the two- and three-body continuum states. The enhancement given by the three-body continuum states is found to have a strong connection to the halo struc…
All possible dissociation cross sections for the loosely bound three-body halo nuclei $^6$He (n+n+$α$) and $^{11}$Li (n+n+$^{9}$Li) are computed as functions of target and beam energy. Both Coulomb and nuclear interactions are included in the same theoretical framework. The measurements agree with the calculations for energies above 100 Mev/nucleon. The largest cross sections correspond to final states with zero or three particles for heavy and with two neutrons for light targets.
Calculations in ab initio no-core configuration interaction (NCCI) approaches, such as the no-core shell model or no-core full configuration methods, have conventionally been carried out using the harmonic-oscillator many-body basis. However, the rapid falloff (Gaussian asymptotics) of the oscillator functions at large radius makes them poorly suited for the description of the asymptotic properties of the nuclear wave function. We establish the foundations for carrying out NCCI calculations with an alternative many-body basis built from Coulomb-Sturmian functions. These provide a complete, dis…
Isospin violation effects in nuclei are investigated within a microscopic approach which takes into account the influence of the residual strong interaction on the Coulomb interaction. The renormalization of the off-diagonal Coulomb field acting within a nucleus, by the residual strong interaction is calculated analytically in a simplified RPA. From the expression for the suppression coefficient of the isospin violating part of the Coulomb field, the isospin violating spreading widths of isobaric analog states are derived. The resulting reduction of the width is in agreement with the data on t…
We propose a novel approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) Faddeev calculations. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. We demonstrate numerically and provide analytical insight that the elastic proton-deuteron (pd) observables can be determined directly from the resulting on shell 3N amplitude increasing the screening radius. The screening limit exists without the need of renormalisation not only for observables but for the e…
We present exact Hartree-Fock-Bogoliubov calculations with the finite range density dependent Gogny force using a triaxial basis. For the first time, all contributions to the Pairing and Fock Fields arising from the Gogny and Coulomb interactions as well as the two-body correction of the kinetic energy have been calculated in this basis. We analyze the relevance of these terms in different regions of the periodic table at zero and high angular momentum. The validity of commonly used approximations that neglect different terms in the variational equations is also checked. We find a decrease of …
Coulomb and nuclear dissociation of $^{17}$Ne on light and heavy targets are studied theoretically. The dipole E1 strength function is determined in a broad energy range including energies of astrophysical interest. Dependence of the strength function on different parameters of the $^{17}$Ne ground state structure and continuum dynamics is analyzed in a three-body model. The discovered dependence plays an important role for studies of the strength functions for the three-body E1 dissociation and radiative capture. The constraints on the $[s^2]/[d^2]$ configuration mixing in $^{17}$Ne and on $p…
The asymptotic wave function derived by Alt and Mukhamedzhanov [Phys. Rev. A 47, 2004 (1993)] and Mukhamedzhanov and Lieber [Phys. Rev. A 54, 3078 (1996)] has been refined in the region where the pair $ (β,γ)$ remains close to each other while the third particle $α$ is far away from them $(ρ_α \to \infty$, $r_α/ρ_α\to 0)$. The improved wave function satisfies the Schrödinger equation up to the terms of order $O(1/{ρ_α}^{3})$, provides the leading asymptotic terms of the three-body scattering wave function with Coulomb interactions and gives further insight into the continuum behavior of the th…
We study the role the proton-electron gas interaction has on the formation of nuclear structures in neutron star crusts. Using a classical molecular dynamics model we study isospin symmetric and asymmetric matter at subsaturation densities and low temperatures varying the Coulomb interaction strength. The effect of such variation is quantified on the fragment size multiplicity, the inter-particle distance, the isospin content of the clusters, the nucleon mobility and cluster persistence, and on the nuclear structure shapes. We find that the proton-electron Coulomb interaction distributes matte…
We accurately analyze the $^6$He+$^{209}$Bi scattering at 19 and 22.5 MeV near the Coulomb barrier energy, using the continuum-discretized coupled-channels method (CDCC) based on the $n$+$n$+$^4$He+$^{209}$Bi four-body model. The three-body breakup continuum of $^6$He is discretized by diagonalizing the internal Hamiltonian of $^6$He in a space spanned by the Gaussian basis functions. The calculated elastic and total reaction cross sections are in good agreement with the experimental data, while the CDCC calculation based on the di-neutron model of $^6$He, i.e., the $^2n$+$^{4}$He+$^{209}$Bi t…
The Fock representation of the Coulomb interaction. II. Three-body bound states
本文在动量空间Faddeev核中引入库仑相互作用的Fock表示，精确计算三体束缚态，验证了$^{3}$H--$^{3}$He劈裂及轻核团簇态的库仑位移。
Three-nucleon scattering equations with irreducible three-nucleon force are solved in momentum-space. The Coulomb interaction between the two protons is included using the method of screening and renormalization. The need for the renormalization of the scattering amplitudes is demonstrated numerically. The Coulomb and Urbana IX three-nucleon force effects on the observables of elastic proton-deuteron scattering and breakup are studied.
Breakup cross sections are determined for the Borromean nucleus 22C by using a four-body eikonal model, including Coulomb corrections. Bound and continuum states are constructed within a 20C + n + n three-body model in hyperspherical coordinates. We compute continuum states with the correct asymptotic behavior through the R-matrix method. For the n+ n potential, we use the Minnesota interaction. As there is no precise experimental information on 21C, we define different parameter sets for the 20C + n potentials. These parameter sets provide different scattering lengths, and resonance energies …
On the basis of the Faddeev integral equations method and the Watson- Feshbach concept of the effective (optical) interaction potential, the first fully consistent three-body approach to the description of the penetration of a charged particle through the Coulomb field of a two-particle bound complex (composed of one charged and one neutral particles) has been developed. A general formalism has been elaborated and on its basis, to a first approximation in the Sommerfeld parameter, the influence of the nuclear structure on the probability of the penetration of a charged particle (the muon, the …
A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the $e^++H$ system both below and above the $H(n=2)$ threshold. We found excellent agreements with previous calculations in most cases.
The Coulomb-nuclear interference is studied as a function of the projectile ground-state binding energy in the 8Li + 12C and 8Li + 208Pb breakup reactions, by considering an arbitrary range for the 8Li ground-state binding energies εb, varying from the experimental one 2.03 MeV down to 0.01 MeV. Regardless the target mass, we first show that the Coulomb breakup cross section is stronger dependent on εb than the nuclear breakup cross section, due to the long-range nature of the Coulomb forces and to the electromagnetic transition matrix elements. For example, in case of 8Li + 208Pb reaction at …
Explicit expressions of the vertex constant for the decay of a nucleus into two charged particles for an arbitrary orbital momentum $l$ are derived for the standard expansion of the effective-range function $K_l(k^2)$, as well as when the function $K_0(k^2)$ has a pole. As physical examples, we consider the bound state of the nucleus ${}^3\rm{He}$ and the resonant states of the nuclei ${^2}$He and ${^3}$He in the s-wave, and those of ${}^5\rm{He}$ and ${}^5\rm{Li}$ in the p-wave. For the systems $Np$ and $Nd$ the pole trajectories are constructed in the complex planes of the momentum and of th…
Coulomb and charge-symmetry breaking effects in the $^3$He ground state within the dressed dibaryon model developed recently for $2N$ and $3N$ forces are examined in detail. Particular attention has been paid to the Coulomb displacement energy $ΔE_C=E_B(^3{\rm H})- E_B(^3{\rm He})$ and rms charge radii of $^3$H and $^3$He. A new scalar $3N$ force between the third nucleon a$ dibaryon is found to be very important for a correct description of the Coulomb energy and rms charge radius in $^3$He. In view of the new results for $ΔE_C$ obtained here, the role of the effects of charge symmetry breaki…
We investigate the three-body Coulomb breakup of a two-neutron halo nucleus $^{11}$Li. We use the coupled-channel $^9$Li + $n$ + $n$ three-body model, which includes the coupling between last neutron states and the various $2p$-$2h$ configurations in $^9$Li due to the tensor and pairing correlations. The three-body scattering states of $^{11}$Li are described by using the combined methods of the complex scaling and the Lippmann-Schwinger equation. The calculated breakup cross section successfully reproduces the experiments. The large mixing of the s-state in the halo ground state of $^{11}$Li …
A novel method for calculating resonances in three-body Coulombic systems is presented. The Faddeev-Merkuriev integral equations are solved by applying the Coulomb-Sturmian separable expansion method. To show the power of the method we calculate resonances of the three-$α$ and the $H^-$ systems.
Recent advances in the treatment of scattering of charged composite particles are reviewed. In a first part I report on developments of the theory. Specifically I describe the recent completion of the derivation of the co-ordinate space asymptotic behaviour of the wave function for three charged particles in the continuum. This knowledge is increasingly being made use of in attempts to `derive' three-Coulomb particle wave functions to be used in all of configuration space which are solutions of the Schroedinger equation, though not everywhere but at least in one or preferably all of the asympt…
We propose a new method to describe three-body breakups of nuclei, in which the Lippmann-Schwinger equation is solved combining with the complex scaling method. The complex-scaled solutions of the Lippmann-Schwinger equation (CSLS) enables us to treat boundary conditions of many-body open channels correctly and to describe a many-body breakup amplitude from the ground state. The Coulomb breakup cross section from the 6He ground state into 4He+n+n three-body decaying states as a function of the total excitation energy is calculated by using CSLS, and the result well reproduces the experimental …
The set of Faddeev and Lippmann--Schwinger integral equations for three-body systems involving Coulomb interactions deduced from a ``three-potential'' picture are shown to be compact for all energies and a method of solution is given.
A practical method to solve cut-off Coulomb problems of two-cluster systems in the momentum space is given. When a sharply cut-off Coulomb force with a cut-off radius $ρ$ is introduced at the level of constituent particles, two-cluster direct potential of the Coulomb force becomes in general a local screened Coulomb potential. The asymptotic Hamiltonian yields two types of asymptotic waves; one is an approximate Coulomb wave with $ρ$ in the middle-range region, and the other a free (no-Coulomb) wave in the longest-range region. The constant Wronskians of this Hamiltonian can be calculated in e…
Three-body charge transfer reactions with Coulomb interaction in the final state are considered in the framework of coordinate-space integro-differential Faddeev-Hahn-type equations within two- and six-state close coupling approximations. The method is employed to study direct muon transfer in low-energy collisions of the muonic hydrogen H$_μ$ by helium (He$^{++}$) and lithium (Li$^{+++}$) nuclei. The experimentally observed isotopic dependence is reproduced.
In this paper, I prove that the sub-Coulomb $A(d,p)B$ reaction amplitude, which is a solution of the three-body Faddeev equations in the Alt-Grassberger-Sandhas (AGS) form, is peripheral if a peripheral is the corresponding DWBA amplitude. Hence the Faddeev's reaction amplitude for the sub-Coulomb $A(d,p)B$ reactions can also be parametrized in terms of the ANC of the $(n\,A)$ bound state. First, I consider the original AGS equations with separable potentials and prove that such equations are peripheral at sub-Coulomb energies. After that, the two-particle AGS equations are derived for the gen…
Theoretical description of reactions in general, and the theory for $(d,p)$ reactions, in particular, needs to advance into the new century. Here deuteron stripping processes off a target nucleus consisting of ${A}$ nucleons are treated within the framework of the few-body integral equations theory. The generalized Faddeev equations in the AGS form, which take into account the target excitations, with realistic optical potentials provide the most advanced and complete description of the deuteron stripping. The main problem in practical application of such equations is the screening of the Coul…
The repulsive Coulomb force poses severe challenges when describing $(d, p)$ reactions for highly charged nuclei as a three-body problem. Casting Faddeev-AGS equations in a Coulomb basis avoids introducing screening of the Coulomb force. However, momentum space partial-wave $t$-matrix elements need to be evaluated in this basis. When those $t$-matrices are separable, the evaluation requires the folding of a form factor, depending on one momentum variable, with a momentum space partial-wave Coulomb function, which has a singular behavior at the external momentum $q$. We developed an improved re…
We investigate effects of the Coulomb force on the nuclear pairing properties by performing the Gogny Hartree-Fock-Bogolyubov calculations for the $N=20$, $28$, $50$, $82$ and $126$ nuclei. The Coulomb force reduces the proton pair energy and the even-odd mass difference by about 25%, except for nuclei at and around the proton shell or subshell closure. We then propose a renormalization scheme via a reduction factor $γ_p$ for the proton pairing channel. It is found that a single value of $γ_p(=0.90)$ well takes account of the Coulombic effect, for nuclei covering wide range of the mass number …
A relationship between the Coulomb inclusive break-up probability and the radiative capture reaction rate for weakly-bound three-body systems is established. This direct link provides a robust procedure to estimate the reaction rate for nuclei of astrophysical interest by measuring inclusive break-up processes at different energies and angles. This might be an advantageous alternative to the determination of reaction rates from the measurement of $B(E1)$ distributions through exclusive Coulomb break-up experiments. In addition, it provides a reference to assess the validity of different theore…
We investigate the Coulomb breakup of the two-neutron halo nucleus 6He by extending the framework of the finite range post form distorted wave Born approximation theory introduced earlier for the description of the breakup of one-neutron halo nuclei. The model can use the realistic three-body wave function for the 6He ground state. Calculations have been performed for energy, total and parallel momentum distributions of 4He fragment. The two-body 4He-neutron and neutron-neutron correlations have also been calculated. Our results are compared with those of an adiabatic model of the Coulomb brea…
For solving the $2\to 2,3$ three-body Coulomb scattering problem the Faddeev-Merkuriev integral equations in discrete Hilbert-space basis representation are considered. It is shown that as far as scattering amplitudes are considered the error caused by truncating the basis can be made arbitrarily small. By this truncation also the Coulomb Green's operator is confined onto the two-body sector of the three-body configuration space and in leading order can be constructed with the help of convolution integrals of two-body Green's operators. For performing the convolution integral an integration co…
The two-body Coulomb scattering problem is solved using the standard complex scaling method. The explicit enforcement of the scattering boundary condition is avoided. Splitting of the scattering wave function based on the Coulomb modified plane wave is considered. This decomposition leads a three-dimensional Schrödinger equation with source term. Partial wave expansion is carried out and the asymptotic form of the solution is determined. This splitting does not lead to simplification of the scattering boundary condition if complex scaling is invoked. A new splitting carried out only on partial…
Scattering of two spinless charge particles for simple forces including coulomb admixtures is calculated without partial wave decomposition. The coulomb interaction being taken is of the type of screened coulomb potential. For the forces range are not infinite, the standard scattering theory is applied. The differential and total cross section is calculated and coulomb effects are shown.
We investigate the 9Be + 208Pb elastic scattering and fusion at energies around the Coulomb barrier. The $^9$Be nucleus is described in a α+ α+ n three-body model, using the hyperspherical coordinate method. The scattering with $^{208}$Pb is then studied with the Continuum Discretized Coupled Channel (CDCC) method, where the α+ α+ n continuum is approximated by a discrete number of pseudostates. Optical potentials for the $α+^{208}$Pb and $n+^{208}$Pb systems are taken from the literature. We present elastic-scattering and fusion cross sections at different energies, and investigate the conver…
The Born approximation, one photon exchange, used for DIS is subject to virtual radiative corrections which are related to the long-range Coulomb forces. They may be sizeable for heavy nuclei since Zαis not a small parameter. So far these corrections are known only for two processes, elastic scattering and bremsstrahlung on the Coulomb field of a point-like target. While the former amplitude acquires only a phase, in the latter case the cross section is modified also. Although the problem of Coulomb corrections for DIS on nuclei is extremely difficult, it should be challenged rather than 'swep…
We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a {\it non-empirical} pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial-wave…
Completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is the basic ingredient of quantum mechanics, plays an important role in nuclear reaction and nuclear structure theory. However, until now, there was no a formal proof of the completeness of the eigenfunctions of the two-body Hamiltonian with the Coulomb interaction. Here we present the first formal proof of the completeness of the two-body Coulomb scattering wave functions for repulsive unscreened Coulomb potential. To prove the completeness we use the Newton's method [R. Newton, J. Math Phys., 1, 319 (1960)]. The proof a…
We present a method for treatment of three charged particles. The proposed method has universal character and is applicable both for bound and continuum states. A finite rank approximation is used for Coulomb potential in three-body system Hamiltonian, that results in a system of one-dimensional coupled integral equations. Preliminary numerical results for three-body atomic and molecular systems like $H^{-}$, He, $ppμ$ and other are presented.
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