An numerical approach for finite volume three-body interaction

In present work, we study an numerical approach to one dimensional finite volume three-body interaction, the method is demonstrated by considering a toy model of three spinless particles interacting with pair-wise $\delta$-function potentials. The numerical results are compared with the exact solutions of three spinless bosons interaction when strength of short-range interactions are set equal for all pairs.

In addition to its application in nuclear/hadron physics, the study of three or more particles either in free space or interacting with periodic potentials also has wide applications and interests in condense matter physics. For an example, the rapid development of semiconductor technology has allowed to manufacture quantum dots (QDs) nanometer sized islands, and these new nano-structure materials have triggered a great interest from both experimental and theoretical points of view [55]. Especially, electrons inside these nano-structures can be controlled experimentally, so it may potentially be applied to the development of materials in quantum * Electronic address: pguo@jlab.org computing [56] and spintronics [57]. Moreover, the QDs are considered as ideal nano-laboratories to study the physical properties of few-particle system in reduced dimensional space. In this regard, two-electron system [58][59][60][61][62][63][64][65][66][67][68] becomes the simplest arrangement of few-particle systems, which may be served as a starting point to evaluate the correlation effects on the energy band structure of more complicated systems. Some simplest and exactly solvable models of one quantum dot with two interacting electrons have been studied in the past, e.g. [69], in which the hybridization effects with the states on the leads are also considered. These early studies show that the effect of the electron-electron interaction, in contrast to the case of non-interacting electrons, indeed changes the electron density of states at the Fermi level, and results in non-trivial corrections to the conductivity and the negative magnetoresistance in disordered conductors [70]. Another class of systems, to which the three and more electrons interaction applies and which is of special interest in condense matter physics, is a many body localization phenomena. In these phenomena, the many-body eigenstates of the Hamiltonian are localized, and Anderson localization type of behavior can only be described by the interacting few-particle dynamics (see, e.g. [71]).
Many attempts on finite volume three-body interactions have been made in recent years [72][73][74][75][76][77][78][79][80][81][82][83] from different approaches. For instance, quantum field theory based diagrammatic approaches or Faddeev equations based method [72][73][74][75][76][77][78][79], and the approach by considering the asymptotic form of wave function in configuration space [82,83]. Unfortunately, majority of these developments are still mathematically unfriendly to common users and are not easily tested in practice because of complication of three-body dynamics. Only a few limited cases of three-body problem can be solved analytically in low dimension, such as McGuire's model in finite volume [82]. However, diffraction effects in McGuire's model are all cancelled out [84], thus no new momenta are created over scattering process, though momenta are allowed to be rearranged among three particles. As the consequence, asymptotic form of wave function contains only plane waves, the spherical waves are completely absent due to the cancellation of diffraction effect [83]. The absence of spherical wave simplifies the algebra of finite arXiv:1709.08255v1 [hep-lat] 24 Sep 2017 volume three-body dynamics dramatically, make it possible to finally have the quantization conditions expressed in quite a simple way analytically [82]. In general cases, the analytic solutions of three-body dynamics are usually not available, even for some seems like simple cases, such as the pair-wise δ-function potentials with unequal strength among pairs [85]. Although, as suggested in [83], given the asymptotic form of wave function, it may be possible to obtain three-body quantization conditions in an analytic form that involves only on-shell scattering amplitudes, obtaining analytic asymptotic form of threebody wave function or parametrization of on-shell threebody scattering amplitudes never is an easy task even for "simple cases", such as unequal strength δ-function pair-wise interactions. Therefore, in present work, we aim to obtain an numerical approach to finite volume three-body problem. Although the explicit and analytic form of quantization conditions are sacrificed and abandoned this way, finite volume three-body problems can be solved numerically and quite reliably without any approximation, and most importantly, the approach is applicable to general cases even when the three-body forces are included. To demonstrate the approach, in this work, we consider a simple toy model of three spinless particles interacting with pair-wise δ-function potentials. The exact solutions in finite volume are available at the limit of equal strength δ-function potentials [82], which can be used to test our numerical approach. As will be made clear later on, the three-body dynamics is completely determined by Faddeev equations, and wave functions in both free space and finite box can be constructed from the solutions of Faddeev equations. The role of matching condition of free space and finite volume wave functions is to impose the extra constraints on allowed energy spectra in a finite box and eventually leads to discrete values of energy spectra as the consequence of periodic lattice structure. In this work, instead of aiming to obtain analytic expressions of three-body quantization conditions which may be derived from the matching condition, we propose to search allowed energy spectra numerically by using matching condition directly. Since three-body dynamics is solely determined by Faddeev equations, and is independent of lattice structure of finite box, the Faddeev equations can thus be solved separately by numerical approach, and solutions may be tabulated and stored regardless the scattering of particles in free space or finite volume. Then, the solutions of Faddeev equations may be used as input into matching condition of finite volume problem to search for allowed discrete energy spectra in a finite box. The strategy of numerical approach is illustrated and tested by a toy model with particles interacting by pair-wise δ-function potentials, the toy model is solved numerically and the results are compared with the exact solutions at the limit of equal strength δ-function potentials among all pairs. At last, we would also like to point out that though our discussion and presentation for finite volume three-body problem has been focused on pair-wise short-range interactions, the approach can be applied to three-body problems in general when threebody force is also included, and the strategy of solving finite volume three-body problem in general cases remains same. A brief discussion of three-body problem by including three-body force is presented in Appendix B. For completeness, a short review of Faddeev's approach for pair-wise short-range interaction is also provided in Appendix A.
The paper is organized as follows. In Section II we summarize the formalism of three-particle interaction in finite volume. The numerical approach and results are presented in Section III. The summary and discussion are given in Section IV.

II. THREE-BODY INTERACTION FOR SHORT RANGE INTERACTION
A. Three-body interaction in free space In this work, for the purpose of demonstration of numerical approach, we consider a non-relativistic toy model of three-body interaction in one spatial dimension, by assuming all particles are spinless and have equal mass. These assumptions are not essential for physics that we are interested in but only to simplify the algebra and presentation. The interactions among particles are assumed pair-wise and described by δ-function potentials with strength, V αβ , between α-th and β-th particles. The three-particle wave function satisfies Schrödinger equation, The three-body problem with pair-wise interactions in free space can be handled by well-known Faddeev's approach [12,13]. In this way, the scattering with either free-three-particle or two-body bound state plus third particle in both initial and final states is treated in the same framework. The details of complete derivations of Faddeev's approach for both scattering of free-threeparticle and scattering on a bound state are listed in Appendix A for the completeness of presentation. A brief discussion for three-body problems with three-body forces is also provided in Appendix B. Hence, only some key results and equations are presented in this section. As proposed in [12,13], the three-body wave function has the form of Ψ = Ψ (0) + 3 γ=1 Ψ (γ) if initial state is freethree-particle, and Ψ = 3 γ=1 Ψ (γ) for scattering of third particle on a two-body bound state. The relative wave functions, e −iP R Ψ (γ) = ψ (γ) (r αβ , r γ ; q ij , q k ), are deter-mined by where t αβ (k) = − mV αβ 2k+imV αβ is the two-body scattering amplitude in pair (αβ), and σ 2 = mE − P 2 (γ) is associated to the incoming wave, if initial state is free-three-particle state, it is given by If the initial state is incident of i-th particle on a bound state of pair (jk), it is thus given by where φ B (γ) (r αβ ) = − mV jk 2 e mV jk 2 |r αβ | refers to the twobody bound state wave function in pair (jk), and 2 ) 2 . In either case, the g (γ) amplitudes satisfy Faddeev type integral equations. In a matrix form, the integral equations for g (γ) amplitudes are given by where G and G (0) stand for column vectors (g (3) , g (1) , g (2) ) T and (g (3) , g (1) , g (2) ) T respectively. The matrix K is given by Eq.(A24), (γ) 's are defined by incoming waves, for an incoming wave of free-three-particle, we have and for scattering of i-th particle by a bound state in pair (jk), thus, Faddeev type equations, Eq. (5), have no analytic solutions due to diffraction effects in general, except the special case when the strengths of δ-function potential among all pairs are identical: [82]. Nevertheless, Eq.(5) can be solved numerically rather straightforwardly in general cases, and the numerical solutions of g (γ) amplitudes can thus be used as input to construct the free space three-body wave function by Eq.(2). As will be presented next, similarly the finite volume three-body wave function is also constructed by using the solutions of g (γ) amplitudes, see Eq.(11).

B. Three-body scattering in finite volume
When particles are confined in a one dimensional periodic box of the size of L, as shown in [82], the relative finite volume wave function must satisfy periodic boundary condition, The finite volume three-body wave function, ψ (L) , can be constructed from three-body free space wave function, ψ, by 3 n βγ L × ψ(r αβ + n αβ L, r k + 1 2 n αβ L + n βγ L; q ij , q k ). (9) The infinite sum may be carried out by using relation Also using the Poisson summation formula, L n) and free space three-body wave function in Eq.(2), the finite volume three-body wave function hence yields The quantization conditions that yield the discrete energy spectra for three-body interaction in a finite box may be obtained by matching condition [82], where both finite volume wave function, ψ (L) (γ) in Eq.(11), and free space wave function, ψ (γ) in Eq. (2), are determined by the solutions of Faddeev equations in Eq.(5). In another word, the three-body dynamics is completely described by Faddeev equations, and the role of quantization conditions or matching conditions in Eq.(12) is nothing but to impose constraints on allowed energy spectra to meet the requirement of periodic lattice structure in a finite box. Therefore, given the solution of Faddeev equations of g-amplitude, the task is to scan all the possible combination of (q ij , q k ) to find the solution of energy spectrum that fulfill the matching condition in Eq. (12) for an arbitrary pair of (r αβ , r γ ). Bearing this in mind, instead of finding the basis of asymptotic form of wave function in both free space and finite volume [41] and deriving an analytic expression of secular equation from matching conditions [82,83], our strategy is to solve Faddeev equations first, and use the solutions of g (γ)amplitudes as input of matching condition, Eq. (12), to search all possible allowed energy spectra of three-body interaction numerically. Although, it seems like the analytic forms of secular equation is lost this way, the numerical approach presented in this work is rather straightforward, and the formalism itself is rather simple and user friendly. The only trade-off is that Faddeev equations has to be solved first numerically, and therefore it is more computationally involved. Fortunately, solving Faddeev equations and searching allowed energy spectra in matching conditions are two independent processes, they can be carried out separately. Therefore in practice, it may be plausible to solve Faddeev equations first for multiple initial momenta and energies, and then proceed with second step of energy spectra searching by using matching condition in Eq. (12). In addition, the procedure and strategy of solving finite volume three-body problem is not limited to only pair-wise interactions, but also could be applied to general cases with three-body forces, see discussion in Appendix B. The idea is demonstrated and compared to exact solutions in next section.

A. Scattering of three-boson in general
Let's consider a totally symmetric free-three-particle incoming wave, At the limit of V 12 = V 23 = V 31 = V 0 , it may describe the scattering of three spinless bosons. Hence, we obtain Normally, it is more stable numerically to separate the δfunction type singular terms by redefining g's amplitudes, where similar to equations of g (γ) 's, integral equations for g (γ) 's are given by, whereĜ andĜ (0) stand for column vectors (2) ) T respectively, andĝ In terms ofĝ (γ) amplitudes, the totally symmetric free space and finite volume wave functions are determined respectively by, and separating the δ-function type singular terms from g's by Eq. (15) has no effects on non-trivial solutions of three-body problem in finite volume, thus for non-trivial solutions, finite volume wave function has the similar form as in Eq. (11), As discussed in previous section, when strength of potentials are not identical, Faddeev equations have no analytic solutions, and have to be solved numerically. Then solutions ofĝ (γ) amplitudes by solving Eq.(16) equations can be used as input to construct both free space and finite volume wave function according to Eq.(18) and Eq. (19). Finally, the discrete spectra of three-body interaction in finite volume may be searched numerically by using matching condition, Eq.(12). In the case of equal strength of δ-function potentials, V 12 = V 23 = V 31 = V 0 , the three-body interaction in finite volume is exactly solvable [82]. For totally symmetric incoming wave, see Eq.(13), the exact solutions of g-amplitude are where t(q) = − mV0 2q+imV0 refers to two-body scattering amplitude.
The totally symmetric wave function is expressed in terms of a single independent coefficient, see [82], ψ sym (r 12 , r 3 ; q ij , q k ) = A sym (r 12 , r 3 )e iq12r12 + A sym (−r 12 , r 3 )e −iq12r12 e iq3r3 + A sym (r 31 , r 2 )e iq23r12 + A sym (−r 23 , r 1 )e −iq23r12 e iq1r3 where r 23 = − r12 2 + r 3 and r 31 = − r12 2 − r 3 , and and We remark that the term ψ sym (0) in Eq.(34) presented in [82] was a typo, see Eq.(21) for the correct version above. The totally symmetric finite volume wave function has the same structure as free space wave function given in Eq. (21), the coefficient in finite volume is given by As discussed in [82], the quantization conditions are obtained by considering the matching condition between free space wave function and finite volume wave function, and for the case of equal strength δ-function potentials, quantization conditions are given in simple forms, where two-body phase shift is given by φ(q) = cot −1 − 2q mV0 .
C. Strategy of searching allowed energy spectra The discrete energy spectra in finite volume are determined by the matching condition of wave functions in free space and finite volume, such as Eq. (12). Therefore, in principle, the task of obtaining three-body energy spectra in finite volume is thus to search all possible combination of (q ij , q k ), so that, the matching condition, Eq.(12), is satisfied for an arbitrary (r 12 , r 3 ).
Normally, in order to explicitly removing (r 12 , r 3 ) dependence in matching condition, the quantization conditions may be further derived by expanding the wave functions in terms of certain orthogonal basis, see [83]. For example, the choice of basis may be made based on the asymptotic behavior of three-body wave functions [83], such as Bessel functions, {J J (σr), N J (σr)}, and e iJθ in (r 12 , r 3 ) plane, where (r, θ) are the radius and polar angle of coordinate, (r 12 , r 3 ), respectively. Therefore, according to asymptotic behaviors of three-body wave function, the wave functions in free space and finite volume nor- mally have the forms, see [83], Hence, the matching condition, ψ = ψ (L) , leads to the quantization conditions that are given by determinant condition in terms of expansion coefficients alone, Unfortunately, except a few special cases, such as, equal strength δ-function potentials, the quantization conditions usually do not possess an simple analytic expression and appear messy and complicated. Moreover, the expansion has to be truncated in practice to solve determinant condition, thus, the convergence of expansion somehow more or less depends on the choice of expansion basis of wave functions. Therefore, instead of making efforts on obtaining quantization conditions and solving determinant condition, such as in Eq.(28), we propose to solve the matching condition directly. Though, the explicit expression of quantization conditions is sacrificed, the procedure and strategy of obtaining discrete energy spectra is actually more clear and straightforward. First of all, the Faddeev equations that define the dynamics of three-body interaction, such as in Eq.(5), are solved numerically. Next, the solutions of Faddeev equations are used as input to construct both free space and finite volume wave functions, see Eq.(2) and Eq. (11). At last, the discrete energy spectra that are determined by periodic lattice structure are obtained by searching matching condition, ψ = ψ (L) , directly.
In this subsection, using scattering of three-boson at the limit of equal strength δ-function potentials as an numerical test, we solve Faddeev equations presented in subsection III A, and use the solution of g-amplitude as input to construct wave functions and further seek the discrete spectra that satisfy the matching condition in Eq. (12). The numerical results are compared with exact solutions. The comparison of the numerical solutions of g-amplitudes, free space wave function and finite volume wave function with exact solutions are presented in Fig.1-3 respectively. The matching condition, ψ = ψ (L) , is solved numerically by root finding method, more specifi- The comparison of numerical solution of center of mass (P = 0) finite volume wave function, ψ sym(L) , constructed by using the solution of Faddeev equations and Eq. (19) with exact solutions given by Eq. (24). The real and imaginary parts of finite volume wave function are presented in upper and lower panels respectively. The colors coding, line styles and parameters are the same as in Fig.2. The finite volume wave function as function of (r, θ) is plotted with a fixed r = 5.5.
cally, a function, M(q 12 , q 3 ), is introduced, ψ(r 12 , r 3 ; q 12 , q 3 ) − ψ (L) (r 12 , r 3 ; q 12 , q 3 ) |ψ(r 12 , r 3 ; q 12 , q 3 )| , where sum of (r 12 , r 3 ) are carried out by choosing some discrete values, for example, in present work, (r 12 , r 3 ) space is discretized in terms of polar coordinate, (r, θ). About 30 points of r values in range r ∈ [1,5] and 100 points of θ ∈ [0, 2π] are taken in the sum. N refers to the total numbers of discrete points of (r 12 , r 3 ) in the sum. For the non-trivial solutions of three-body energy spectra (none of particle momentum coincides with 2π L n, n ∈ Z), the possible discrete values of pair (q 12 , q 3 ) are searched by performing root finding of condition, M(q 12 , q 3 ) = 0. The results are compared with the solutions given by quantization condition in Eq. (25), and presented in Fig.4. We remark that the solutions of quantization condition for q ij = 0 are excluded in Fig.4, it is not difficult to see that the wave functions vanish due to the symmetry of three-body for q ij = 0 at the limit of V 12 = V 23 = V 31 = V 0 , thus, the solutions for q ij = 0 are considered as trivial and not included in Fig.4.
The colored lines are labeled by pari of integers, (n1, n2), which are associated to momenta of non-interacting particles by relations: p1,2,3 = 2π L n1,2,3. As examples of both discrete energy and momenta in finite box, the three-body energy levels for L = 20 are also labeled by pair of discrete values of (q12, q3) next to corresponding energy level.

IV. DISCUSSION AND CONCLUSION
As mentioned in previous sections, the dynamics of three-body interaction is completely determined by Faddeev equations regardless the three-particle interacting in free space or finite box. When the periodic boundary condition is considered, the allowed energy spectra are constrained by matching condition between free space three-body wave function and finite volume wave function, and eventually become discrete. Therefore, seeking discrete three-body energy spectra in finite box can be carried out by two independent procedures, first of all, solving dynamical Faddeev equations and using the solutions of Faddeev equations to construct wave functions. Secondly, in stead of seeking an analytic expression of quantization conditions for finite volume three-body interaction that may be derived from matching condition of three-body wave functions, we propose to search for discrete energy spectra by using matching condition directly.
In this work, we demonstrated this approach by considering scattering of three spinless bosons interacting with δ-function potentials. In this case, the exact analytic so-lutions exist at the limit of equal strength of δ-function potential among all pairs. The finite volume three-body interaction in this toy model is then solved numerically, and the discrete energy spectra are searched by using matching condition of wave functions. Finally, all the numerical results are compared with exact solutions presented in section III. At last, we want to stress that although a specific toy model with only pair-wise interaction is solved in this work, this approach is in fact not limited to only pair-wise interaction. The strategy and procedure is applicable to the more general cases when three-body forces are involved, due to the fact that the three-body dynamics and constraints on allowed energy spectra by periodic lattice structure are two independent procedures and can be carried out separately.

V. ACKNOWLEDGMENTS
We acknowledge support from Department of Physics and Engineering, California State University, Bakersfield, CA.

Appendix A: Three-body interaction and Faddeev equations
In this section, we consider scattering of three spinless particles of equal masses, interacting by δ-function potentials of strength, V αβ , between α-th and β-th particles. The three-particle wave function satisfies Schrödinger equation, where m, p i (i = 1, 2, 3) and E = 3 i=1 p 2 i 2m refer to the mass of particle, particle's inital momenta and three-body total energy respectively. As shown in [82], the center of mass, relative positions and corresponding conjugate momenta among particles are defined by R = x1+x2+x3 respectively. Because of translational invariance, the center of mass motion is described by a plane wave, the total three particles wave function is given by, Ψ(x 1 , x 2 , x 3 ; p 1 , p 2 , p 3 ) = e iP R ψ(r αβ , r γ ; q ij , q k ), where ψ(r αβ , r γ ; q ij , q k ) describes relative motions of three particles.

Scattering of three free particles
With an incoming wave of three free particles state, Ψ (0) , three-body wave function has the form [12,13], As shown in [82], the Lippmann-Schwinger equation for relative wave function, ψ (γ) , can be obtained as where t αβ (k) = − mV αβ 2k+imV αβ refers to the two-body scattering amplitude between α-th and β-th particles for a δ-function potential interaction, and σ 2 = mE − P 2 6 = q 2 ij + 3 4 q 2 k . Instead of solving Faddeev T -matrix equations, see [82], numerically, it is more convenient to introduce amplitudes, g (γ) , by The relative wave function, ψ (γ) , thus can be written as Using Eq.(A4) and Eq.(A5), the integral equations for g (γ) -amplitude are obtained, where g (γ) is given by, The g (γ) -amplitude is related to standard Faddeev Tamplitude by the full three-body scattering amplitude is given by where k α = −k αβ − kγ 2 and k β = k αβ − kγ 2 .

Scattering on a two-body bound state
For the case of scattering on a bound state, e.g. i-th particle incident on a bound state of (jk) pair, threebody wave function thus has the form of Ψ = 3 γ=1 Ψ (γ) , and the Lippmann-Schwinger equation for relative wave function reads, where the bound state wave function in pair (jk) |r αβ | , and 2 ) 2 refers to the relative momentum between incident i-th particle and pair (jk) bound state. The integral equations for g (γ) -amplitude for scattering on a bound state also has the same form of Eq.(A6), where for the case of scattering on a bound state in (jk) pair, g (γ) is given by 3. Exact solutions for equal strength of δ-function potentials: V12 = V23 = V31 = V0 Only for the special case with equal strength of δpotential among all pairs, V 12 = V 23 = V 31 = V 0 , threebody interactions are in fact exactly solvable [82,86,87]. The exact solutions can be used for testing and verifying numerical approach, and also for completeness, the exact solutions of Faddeev equations, Eq.(A6), are presented in follows for scattering of three free particles with incoming wave e iq12r12 e iq3r3 and for scattering on a bound state of pair (12) respectively. a. Scattering of three free particles with incoming wave e iq 12 r 12 e iq 3 r 3 For incoming wave e iq12r12 e iq3r3 , g (γ) are thus given by (2) (k; q ij , q k ) = −i where t(q) = − mV0 2q+imV0 . The exact solutions of gamplitude are Scattering on a bound state of pair (12) For incoming wave φ B (3) (r 12 )e iq B 3 r3 , the g (γ) are given by The exact solutions of g-amplitude are (A20) 4. Numerical test for scattering of three free particles with an incoming wave e iq 12 r 12 e iq 3 r 3 In general, Faddeev equations, Eq.(A6), have to be solved numerically. The numerical approach is rather straightforward for the case of scattering on a bound state, the expression of g (0) (γ) (see Eq.(A18)) does not contain δ-function type singularities. Thus Eq.(A6) is standard Fredholm-type integral equation, and can be solved easily by matrix inversion method. The special care has to be given to the case of scattering of three free particles, in this case, g (γ) (see Eq.(A12-A14)) does indeed contain δ-function type singularities. The singularities in Eq.(A6) can be removed by redefining g (γ) 's. For example, given g (3) contains singular term it 23 (−q 23 )2πδ(k − q 3 ), by a shifting in g (3) , the new integral equations forĝ (γ) 's are thus free of δfunction type singularities, and are also Fredholm-type equations. In addition, extra care has to be taken when it comes to the branch cut of square root terms, and pole contributions in Faddeev equations. The pole contributions are handled by using standard i prescription, see Eq.(A6) for instance. As for branch cut contribution, we adopt the same convention as used in [82], for the square root terms, q 2 αβ , we assign a small imaginary part to q 12 → q 12 + i0 + , the imaginary part for q 23 → q 23 − i0 + and q 31 → q 31 − i0 + are thus determined by relations, q 23 = − 1 2 q 12 + 3 4 q 3 and q 31 = − 1 2 q 12 − 3 4 q 3 respectively. In addition, our convention for complex square root is given by As the demonstrations of some numerical tests, the g (γ) 's equations are solved numerically for an incoming wave of e iq12r12 e iq3r3 and compared with the exact solutions presented in section A 3 a. As already mentioned previously, the δ-function type singularities have to be removed by shifting g (γ) 's, we thus obtain integral equations forĝ (γ) 's, whereĜ andĜ (0) stand for column vectors (ĝ (3) ,ĝ (1) ,ĝ (2) ) T and (ĝ (3) ,ĝ (1) ,ĝ (2) ) T respectively, the dependence on initial momenta (q ij , q k ) has been dropped in above equation. The matrix K is given by (1) (k; q ij , q k ) = i (2) (k; q ij , q k ) = −i 2q 12 it 12 (q 12 ) [1 + it 23 (−q 23 )] (k − q 2 − i )(k − q 1 + i ) Faddeev equations forĝ (γ) 's, given by Eq.(A23), are solved numerically by matrix inversion method, and the comparison of numerical solutions with exact solutions in Eqs.(A15-A17) is presented in Fig.5.

Appendix B: Faddeev equations including three-body force
In previous sections, our discussion of three-body problem has been restricted on the interactios of three particles with only pair-wise δ-function potentials. In this section, we would like to extend our discussion of three-body interaction by including a spherical symmetric threebody force potential, U (r) where r = r 2 αβ + 4 3 r 2 γ , and give a brief presentation how the three-body force may be handled in Faddeev equations approach. By including a three-body force potential, U (r), Schrödinger equation now has the form of Let's consider the scattering of three-particle with an incoming wave of three free particles, Ψ (0) , three-body wave function may thus be expressed in the form of where Ψ (γ) satisfies equation, and similarly the equation for Ψ (U ) is given by = −U (r) Ψ (0) + Ψ (α) + Ψ (β) + Ψ (γ) .