Oct 27, 1998 - ... of the two-dimensional Faddeev equations we used a Lanczos-type .... [8] H. Wita la, T. Cornelius, and W. GlÃ¶ckle, Few-Body System...

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arXiv:nucl-th/9810073v1 27 Oct 1998

Institute of Nuclear and Particle Physics, and Department of Physics, Ohio University, Athens, OH 45701

W. Sandhas Physikalisches Institut der Universit¨ at Bonn, Endenicher Allee 11-13, D-53115 Bonn, Germany

J. Haidenbauer Institut f¨ ur Kernphysik, Forschungszentrum J¨ ulich GmbH, D-52425 J¨ ulich, Germany

A. Nogga Institut f¨ ur Theoretische Physik II, Ruhr-Universit¨ at Bochum, D-44780 Bochum, Germany (October 22, 1998)

Abstract 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 j ≤ 2. 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. PACS number(s): 21.60.-n, 27.10.+h

Typeset using REVTEX 1

I. INTRODUCTION

Separable approximations or expansions of the underlying two-body interaction play an essential role in calculations of few-nucleon systems. In the three-body problem such an input reduces the two-dimensional Faddeev equations, or the corresponding Alt-GrassbergerSandhas (AGS) equations, to one-dimensional eﬀective two-body equations [1]. Analogously, the four-body AGS equations go over into eﬀective three-body, and, after repeated application of separable expansions, into eﬀective two-body equations [2]. A considerable reduction of the complexity of the original problem is achieved in this way. What remains to be done, however, is a careful test of accuracy of the respective separable representations. In early calculations Yamaguchi-type separable potentials were employed. Though not very realistic, they simulated characteristic aspects of the nuclear interaction, leading thus to qualitatively correct cross sections for three-nucleon scattering. Stimulated by this experience, rather sophisticated separable approximation or expansion techniques were developed and have been applied successfully to realistic interactions. The Ernst-Shakin-Thaler (EST) expansion [3] is considered as a particularly powerful separable approach. Its eﬃciency has been thoroughly investigated for the Paris potential [4]. Indeed, based on high-rank separable expansions of this type [5,6], rather accurate predictions for the neutron-deuteron (n-d) cross section and some polarization observables were achieved [6,7]. Later on, when direct solutions of the two-dimensional three-body equations became available [8], it was established that the results obtained in this way and with the EST approach are in excellent agreement [9,10]. The W-matrix method [11] provides quite directly a rank-one separable representation of the two-body T matrix, which preserves the analytical properties of the original T matrix. For the Malﬂiet-Tjon (MT I+III) potential three-body bound-state and scattering results [12,13], including break-up [14], were obtained on this basis in complete agreement with alternative treatments. Calculations for the Paris potential were also found to be rather reliable for most, but not all observables [13]. 2

For further applications of separable approximations or expansions, e.g., in processes as the photodisintegration of three-nucleon systems [15,16], the corresponding radiative capture [17], or the pion absorption on such systems [18], additional tests of accuracy of the threebody wave functions involved appear desirable. In case of a 5 channel calculation and the EST method such a test has been made already in Ref. [19] for the Paris potential, and, with a modiﬁed expansion method, in Ref. [20] for the Argonne 14 potential restricted to j ≤ 2. In what follows we compare triton bound-state calculations performed by means of the EST expansion and the W-matrix approximation with direct solutions of the twodimensional homogeneous three-body equations. In view of the sensitivity of the bound-state problem to diﬀerences in the two-body input, this comparison should provide a particularly relevant test of accuracy of these methods. The paper is organized as follows: Sections II and III brieﬂy describe the W-matrix and EST methods, respectively. In Sec. IV we give the relevant equations for the bound-state calculations using separable potentials (details of the direct treatment can be found in Refs. [21,22]). Our results and conclusions are presented in the last section.

II. W-MATRIX REPRESENTATION

The W-matrix method [11,13] leads to a quite appropriate splitting of the two-body T matrix into one dominant separable part and a small non-separable remainder Tll′ (p, p′ ; E + i0) = S Tll′ (p, p′ ; E + i0) + R Tll′ (p, p′ ; E + i0).

(2.1)

Similarly to the T matrix, the W matrix is deﬁned by an equation of Lippmann-Schwinger type, however, with a modiﬁed non-singular kernel. After partial wave decomposition this equation is given by (we use units (¯hc)2 = 2µ = 1 and thus E = p2 ) ∞

Wlηˆl (p, p′ ; E)

=

Ulηˆl (p, p′)

+

XZ l′ 0

Ullη′ (p, q) − Ullη′ (p, k) 2l′ η q Wl′ˆl (q, p′ ; E), dq q 2 E−q 2

3

(2.2)

where k is subject to the constraint k =

√

E for E ≥ 0 and is kept arbitrary for E < 0.

Here l and ˆl are the orbital angular momenta, and η = (s, j; t) stands for the spin, the angular momentum j [ with the coupling sequence (l, s)j], and the isospin t of the two-body subsystem. The input entering Eq. (2.2) is related to the two-body potential matrix Vlˆηl (p, p′) ˆ

according to Ulηˆl (p, p′ ) = p−l Vlηˆl (p, p′ ) p′l . The separable part of the W-matrix representation (2.1) of the two-body T matrix is given by S

Tllη′ (p, p′ ; E + i0) =

X ˆ llˆ′

pl Wlηˆl (p, k; E) ∆ˆηlˆl′ (E + i0) Wlη′ˆl′ (p′ , k; E) p′l

′

(2.3)

and the nonseparable remainder reads R

Tllη′ (p, p′ ; E + i0) = pl Wllη′ (p, p′ ; E) −

X ˆ lˆ l′

Wlηˆl (p, k; E) (Wˆlηˆl′ (k, k; E))−1 Wˆlη′ l′ (p′ , k; E) p′l . ′

(2.4) The propagator ∆ˆηlˆl′ is given by ∆ˆηlˆl′ =

X ˆ l′′

(Fˆlηˆl′′ (E + i0))−1 (Wˆlηˆl′′ (k, k; E))−1 ,

(2.5)

where Fˆlηˆl′′ (E + i0) is a generalization of the Jost function (Fˆlηˆl′ (E

+ i0))

−1

= δˆlˆl′ −

Z∞

ˆ Wˆ lˆ l′ (q, k; E) E + i0 − q 2

dq q 2 q 2l

0

.

(2.6)

On-the-energy-shell and half-oﬀ-shell the separable part (2.3) of (2.1) is identical with √ the exact T matrix. In fact, when inserting the momentum k = E for one or both of the momenta p or p′ , we easily see that the remainder (2.4) vanishes. Therefore, the separable part of the T matrix has the same pole and cut structure as the full T matrix. This suggests to approximate the T matrices, entering the kernel of Faddeev-type equations, by the separable expression (2.3). To optimize this approximation, i.e., to minimize the eﬀect of the neglected remainder, two criteria for the choice of the functional form of the free parameter k were developed [13]. In the present work we apply the criterion based on the Schmidt norm of the kernel of the three-body equations. In the past this method has been used in elastic n-d scattering calculations [12,13], in the breakup case [14], and more recently in the photodisintegration of the triton [15]. 4

III. EST METHOD

The Ernst-Shakin-Thaler (EST) method [3] allows one to generate separable representations of arbitrary rank N that agree exactly (on- and half-oﬀ-shell) with the original T matrix at N speciﬁc, appropriately chosen energies. For a brief outline of this method let us begin with the (partial-wave projected) Lippmann-Schwinger equation for the wave function, |ψE i = |kE i + G0 (E)V |ψE i,

(3.1)

where |kE i is the incoming wave, and G0 (E) the two-body Green’s function. The dependence on the orbital angular momenta l, l′ and on the conserved quantum numbers (the spin, the angular momentum j [ with the coupling sequence (l, s)j], and the isospin t of the two-body subsystem) is suppressed for convenience. For proper scattering solutions (on-shell), kE and E are related by E = kE2 . According to the EST method, a rank-N separable representation of a potential V is given by the form V EST =

N X

µ,ν=1

V |ψEµ i Λµν hψEν |V,

(3.2)

where Eµ , (µ = 1, ..., N), is a freely choosable, but ﬁxed set of energies. The coupling strengths Λµν are determined by the condition N X

ν=1

Λµν hψEν |V |ψEρ i = δµρ .

(3.3)

Note that the ”form factors” in the separable potential (3.2) consist of the objects V |ψEµ i, where |ψEµ i are solutions of Eq. (3.1) at the energies E = Eµ . Thus, Eq. (3.3) together with Eq. (3.2) implies that the following relation holds at the N energies Eµ V EST |ψEµ i = V |ψEµ i = T (Eµ )|kEµ i = T EST (Eµ )|kEµ i.

(3.4)

Here T and T EST are the two-body T matrices for the potential V and its separable representation V EST , respectively. Evidently Eq. (3.4) means that the on-shell, as well as the 5

half-oﬀ-shell T matrices for both interactions, V and V EST , are exactly the same at the energies Eµ . With the form factors |gνη li = V η |ψEη ν il (η = (s, j; t)) and the propagators ∆ηµν of this representation, the two-body T matrix reads Tllη′ (E + i0) =

X µν

|gµη li ∆ηµν (E + i0) hgνη l′ |

(3.5)

−1

(3.6)

where

∆η (E + i0) = (Λη )−1 − G0 (E + i0) and (G0 (E + i0))µν =

X l

hgµ l|G0 (E + i0)|gν li).

(3.7)

For more details of this construction we refer to Refs. [4,5]. The chosen approximation energies Eµ for the interaction models considered in the present study are summarized in Table I. These energies completely specify the separable expansion (3.2) Note that, for reasons of convenience, we have represented the (numerically given) form factors analytically (cf. Eqs. (2.1) and (2.2) of Ref. [6]), and have performed the actual calculations with these expressions. The corresponding parametrizations can be obtained from one of the authors (J.H.) on request. In Sec. V we are going to present three-nucleon bound state results, based on the EST representation, with a varying rank in various two-body partial waves. We characterize these representations by (n1 n2 n3 ...), where n1 , n2 , n3 , ... stand for the ranks in the 1 s0 , 3 s1 − 3 d1 , 1

p1 , 3 p0 , 3 p1 , 1 d2 , 3 d2 , and 3 p2 − 3 f2 NN partial waves. For a speciﬁc rank nµ in a certain

partial wave the approximation energies Eµ can be read oﬀ from Table I. They are given by the ﬁrst nµ entries. The notation in the results of the three-body calculations is done in the same order of the partial waves, but the ranks of the unused partial waves are left out.

6

IV. 3N BOUND–STATE CALCULATION

The triton bound state |Ψt i is determined by the eigenvalue equation (Et − H) |Ψti = 0,

(4.1)

where the total Hamiltonian H is given by H = H0 + V = H0 +

3 P

γ=1

Vγ . Here we have used

the complementary notation Vγ = Vαβ for the two-body potentials, while H0 denotes the free three-body Hamiltonian. When introducing the corresponding resolvent G0 (z) = (z − H0 )−1 or the channel resolvents Gγ (z) = (z − H0 − Vγ )−1 , Eq. (4.1) can be written in form of homogeneous integral equations, |Ψt i = G0 (Et ) V |Ψt i = G0 (Et )

X

|Ψt i = Gγ (Et ) V¯γ |Ψt i = Gγ (Et )

Vγ |Ψt i

(4.2)

γ

X

(1 − δγβ ) Vβ |Ψt i,

(4.3)

β

with V¯γ = V − Vγ being the channel interaction between particle γ and the (αβ) subsystem. The latter equation can also be understood as a representation of the bound state by the “form-factors” |Fγ i = V¯γ |Ψt i, |Ψt i = Gγ (Et ) |Fγ i.

(4.4)

Multiplying this representation with (1 − δβγ )Vγ , using the relation Vγ Gγ = Tγ G0 , and summing over γ, we obtain for |Fβ i the coupled set of homogeneous integral equations |Fβ i =

X γ

(1 − δβγ ) Tγ (Et ) G0 (Et ) |Fγ i.

(4.5)

Note that this relation may alternatively be derived by going to the bound-state poles of the AGS equations, providing thus their homogeneous version [1]. From (4.2) and (4.4) we infer that the solutions of Eq. (4.5) provide |Ψt i according to |Ψt i =

X γ

G0 (Et ) Tγ (Et ) G0 (Et ) |Fγ i =

X γ

|ψγ i.

(4.6)

The |ψγ i are the standard Faddeev components, as seen by using the deﬁnition of |Fγ i and the relation (4.3), 7

|ψγ i = G0 Tγ G0 |Fγ i = G0 Vγ Gγ |Fγ i = G0 Vγ Gγ V¯γ |Ψt i = G0 Vγ |Ψt i.

(4.7)

Equation (4.5) will be treated numerically in momentum space, employing a complete set of partial-waves states |p q l b Γ Ii. The label b denotes the set (ηKL) of quantum numbers, where K and L are the channel spin of the three nucleons [with the coupling sequence (j, 21 )K] and the relative angular momentum between the two-body subsystem and the third particle, respectively. Γ is the total angular momentum following from the coupling sequence (K, L)Γ, and I is the total isospin. These states satisfy the completeness relation 1=

∞ ∞ XZ Z

blΓI 0 0

dp p2 dq q 2 |p q l b Γ Iihp q l b Γ I|.

(4.8)

The required antisymmetry under permutation of two particles in the subsystem can be achieved by choosing only those states which satisfy the condition (−)l+s+t = −1 . Table II contains the quantum numbers of the corresponding channels taken into account. Inserting the separable T matrix (3.5) for the EST potentials and deﬁning ∞

Fβµb (q)

=

XZ l

0

η dp p2 glµ (p) hp q l b ΓI|G0 |Fβ i,

(4.9)

Eq. (4.5) goes over into ∞ µb

F (q) =

XXZ b′

νρ 0

′

′

′

′ η ρb 3 ′2 dq ′ q ′2 AV bb (q ′ ), µν (q, q , Et ) ∆νρ (Et − 4 q ) F

(4.10)

with ′ ′ AV bb µν (q, q , Et )

=2

∞ ∞ XZ Z ll′ 0 0

′

η dp p2 dp′ p′2 glµ (p) hp q l b Γ I|G0(Et )|p′ q ′ l′ b′ Γ Ii glη′ν (p′ ) (4.11)

being the so-called eﬀective potential. The recoupling coeﬃcients entering this equation can be found in Ref. [13] (or in a more compact form in [21] for another coupling sequence which can easily be changed to the present one). In case of the W-matrix representation Eqs. (4.10) and (4.11) are of similar form and, therefore, not given here. After discretization Eq. (4.10) can be treated as a linear eigenvalue problem, where the energy is considered as a parameter which is varied until the corresponding eigenvalue 8

equals unity. The eigenvalues can be found by using standard numerical algorithms. A better approach is an iterative treatment, known as ”power method” [23], which is justiﬁed due to the compactness of the kernel of the integral equation employed. It was found that this method is much faster than standard eigenvalue algorithms and yields the same accuracy. For the direct solution of the two-dimensional Faddeev equations we used a Lanczos-type algorithm [24,25] that is even more eﬃcient. For the integration in Eq. (4.10) a standard Gauss-Legendre mesh was chosen. The angular integration in the eﬀective potential was done with 16 grid points. Table III contains results for the Paris (EST) potential for diﬀerent numbers of partial wave and an increasing number of mesh points for the q integration. In all cases 36 grid points were suﬃcient to get the binding energy up to 5 signiﬁcant ﬁgures. For all further calculations we have used 40 mesh points to get wave functions of high accuracy. In the calculations we also included q = 0 to avoid extrapolations for small momenta in further applications. The same was done for both variables in the calculation of the wave function. The binding energies obtained in EST and W-matrix approximation for diﬀerent potentials are given in Tables IV–VI, compared with results from a two-dimensional treatment [21] of the Faddeev equations. The whole wave function can now be calculated by either using Eq. (4.6), or by applying the permutation operator P on one Faddeev component [21] |Ψt i = (1 + P ) |ψ1 i,

(4.12)

where P represents the sum of all cyclical and anticyclical permutations of the nucleons. From the practical point of view the latter method has to be preferred. Once |ψ1 i is computed from |ψ1 i = G0 (Et ) T1 (Et ) G0 (Et )|F1 i the calculation of the full wave wave function via Eq. (4.12) is independent of the rank of the separable approximation, which considerably reduces the computing time. The wave function is normalized according to hΨt |Ψt i =

X γ

hψγ |Ψt i = 3 hψ1 |Ψt i = 1. 9

(4.13)

It should be noted that, inserting the completeness relation (4.8) into hΨt |Ψt i, one has to deal with an inﬁnite number of states due to the resulting recouplings. In contrast, when inserting the completeness relation in hψ1 |Ψt i, one has to deal with a ﬁnite number of partial waves corresponding to the ones in hψ1 |. Plots of the wave functions for the Paris (EST) and Bonn A (EST) potentials are shown in Figures 1 - 6. The ﬁgures for the Bonn B (EST) potential are not distinguishable by eye from the ones for the Bonn A (EST) potential, and are therefore not shown.

A. Properties of the wave function

It is common to investigate the properties of the wave function in the LS-coupling scheme (for simplicity we skip the dependence on the isospins in the notation, since they are not recoupled) |p q ((lL)L(sS)S)ΓMΓ i.

(4.14)

In this scheme ﬁrst the two orbital angular momenta and the two spins are coupled separately. The total orbital angular momentum is then coupled with the total spin to the total angular momentum of the three-body system. The total angular momentum of the triton is Γ = 1/2, the total spin S of three particles can be S = 1/2 or S = 3/2. The total orbital angular momentum is, therefore, restricted to L = 0, 1, 2. The transformation from the channel spin into the LS coupling scheme is given by h((lL)L, (sS)S)ΓMΓ | =

X jK

ˆ (−)l+s+L+S+L+S+1ˆj LˆSˆK

with the abbreviation ˆj =

√

l

S

S

K

l

S

K

Γ L L j s

h(((ls)jS)KL)ΓMΓ |,

(4.15)

2j + 1 used only in this equation. It should be noted that for

this transformation the wave function (4.12) has to be projected on all states that give a contribution due to Eq. (4.15). Otherwise the normalization constant of the wave function is changed. It is not suﬃcient to use only those channels used in the calculation of the 10

Faddeev component. Here again the recoupling of channels, as in the calculation of the normalization constant, plays a role. In the LS coupling scheme the wave function can be classiﬁed according to the contributions of the states belonging to L = 0, 1, 2. 1 = 3 hψ1 |Ψt i =3

∞ ∞ XX X Z Z L

=

X

S lLsS 0 0

dp p2 dq q 2 hψ1 |p q ((lL)L(sS)S)Γi hp q ((lL)L(Ss)S)Γ|Ψt i

P(L) = P(S) + P(S′ ) + P(P) + P(D).

(4.16)

L

The contributions to the normalization constant for a certain total angular momentum are denoted by P(L). In case of L = 0 also the symmetric and mixed symmetric spatial contributions P(S) and P(S’), are extracted. The antisymmetric part P(S”) of the wave function is negligible and, therefore, has been omitted. The main contribution to the mixed symmetric part stems from the diﬀerence between the 1 s0 and the 3 s1 interaction.

V. RESULTS AND CONCLUSIONS

As a test of accuracy of the EST and W-matrix approaches we compare the triton binding energies and wave functions obtained in this way with the results of a two-dimensional treatment of the Faddeev equations. In Tables IV-VI our binding energies obtained for the Paris, Bonn A, and Bonn B potentials are given for diﬀerent combinations of partial waves. For all three potentials considered here, the EST results converge with increasing rank and agree within 0.2% with the two-dimensional results. In case of the W matrix, the free parameter k was chosen diﬀerently in each partial wave according to the criterion employed in Ref. [13], which consists in providing a binding energy close to the results obtained with other methods. This optimization method is not fully satisfactory in the j ≤ 2 calculations, due to ambiguities in ﬁxing k when including the 3 p2 -3 f2 partial wave. The agreement of the W-matrix calculations with the two-dimensional results is less good than in the EST case. There are diﬀerences between 0 ≤ 4%. 11

Our ﬁve-channel, i.e., j ≤ 1+ , calculations for the Paris (EST) potential are also in perfect agreement with the results by Parke et al. [19]. We can, moreover, compare with the results by Friar et al. [26] generated in coordinate space, and ﬁnd again good agreement. For j ≤ 2 and the Paris potential our results are exactly the same as those of Ref. [10]. The binding energies for Bonn A and j ≤ 2 diﬀer only slightly from those by Fonseca and Lehman [27]. For the Paris (EST), Bonn A (EST), Bonn B (EST) potentials the components of the three-body wave function, deﬁned in the previous section, diﬀer from the two-dimensional results by 0-2.5%, 0-0.05%, and 0-3% respectively. It should be emphasized, however, that only P(P) shows the large deviation of 3% quoted, while all other components are in much better agreement. The W-matrix results for the components of the wave function diﬀer by 0-12%, 0-10%, and 0-10% in case of the Paris, Bonn A, and Bonn B potentials, respectively. Here the largest deviations are found both in the P(P) and in the P(S’) components. Another test is given by the norm squared of the diﬀerences of the triton wave functions obtained in the separable and the two-dimensional treatments, ∆N = ||Ψsep − Ψ2d ||2 . For the EST potentials ∆N is of the order 10−6 , while for the W-matrix representation it is of the order 10−5 . Also in this respect the EST method, hence, leads to better results. Thus, we have demonstrated the high quality of the EST expansion method. For j ≤ 1+ and the Paris (EST) interaction this has been done already in [19], and for the Argonne potential (with a modiﬁed expansion scheme) in [20]. Here we have extended these former investigations up to j ≤ 2, using moreover two versions of the Bonn potential. The accuracy achieved within the W-matrix approach is less satisfactory, namely of the order 2-10%. But it should be recalled that this treatment is based on a rather simple rank-one approximation only.

12

ACKNOWLEDGMENTS

The work of W. S. and W. Sch., and that of A. N. was supported by the Deutsche Forschungsgemeinschaft under Grant Nos. Sa 327/23-1 and GL-8727-1, respectively. Part of this work has been done under the auspices of the U. S. Department of Energy under contract No. DE-FG02-93ER40756 with Ohio University. The numerical calculations were partly performed on the Cray T3E of the H¨ochstleistungsrechenzentrum in J¨ ulich, Germany.

13

REFERENCES [1] E. O. Alt, P. Grassberger, and W. Sandhas, Nucl. Phys. B2, 167 (1967). [2] P. Grassberger and W. Sandhas, Nucl. Phys. B2, 181 (1967). [3] D. J. Ernst, C. M. Shakin, and R. M. Thaler, Phys. Rev. C 8, 46 (1973); Phys. Rev. C 9, 1780 (1974). [4] J. Haidenbauer and W. Plessas, Phys. Rev. C 30, 1822 (1984). [5] J. Haidenbauer and Y. Koike, Phys. Rev. C 34, 1187 (1986). [6] Y. Koike, J. Haidenbauer, and W. Plessas, Phys. Rev. C 35, 396 (1987). [7] Y. Koike and J. Haidenbauer, Nucl. Phys. A463, 365c (1987). [8] H. Witala, T. Cornelius, and W. Gl¨ockle, Few-Body Systems 3, 123 (1988). [9] T. Cornelius, W. Gl¨ockle, J. Haidenbauer, Y. Koike, W. Plessas, and W. Witala, Phys. Rev. C 41, 2538 (1990). [10] S. Nemoto, K. Chmielewski, N. W. Schellingerhout, J. Haidenbauer, S. Oryu, and P. U. Sauer, Few-Body Systems, in press (1998). [11] E. A. Bartnik, H. Haberzettl, and W. Sandhas, Phys. Rev. C 34, 1520 (1986). [12] E. A. Bartnik, H. Haberzettl, T. Januschke, U. Kerwath, and W. Sandhas, Phys. Rev. C 36, 1678 (1987). [13] Th. Januschke, T. N. Frank, W. Sandhas, and H. Haberzettl, Phys. Rev. C 47, 1401 (1993). [14] T. N. Frank, H. Haberzettl, Th. Januschke, U. Kerwath, and W. Sandhas, Phys. Rev. C 38, 1112 (1988). [15] W. Schadow and W. Sandhas, Phys. Rev. C 55, 1074 (1997).

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[16] W. Sandhas, W. Schadow, G. Ellerkmann, L. L. Howell, and S. A. Soﬁanos, Nucl. Phys. A631, 210c (1998); W. Schadow and W. Sandhas, Nucl. Phys. A631, 588c (1998). [17] W. Schadow and W. Sandhas, to appear in Phys. Rev. C. [18] L. Canton and W. Schadow, Phys. Rev. C 56, 1231 (1997); L. Canton, G. Cattapan, G. Pisent, W. Schadow, and J. P. Svenne, Phys. Rev. C 57, 1588 (1998). [19] W. C. Parke, Y. Koike, D. R. Lehman, and L. C. Maximon, Few-Body Systems 11, 89 (1991). [20] Y. Koike, W. C. Parke, L. C. Maximon, and D. R. Lehman, Few-Body Systems 23, 53 (1997). [21] W. Gl¨ockle, The Quantum Mechanical Few-Body Problem (Springer-Verlag, BerlinHeidelberg, 1983). [22] A. Nogga, D. H¨ uber, H. Kamada, and W. Gl¨ockle, Phys. Lett. B 409, 19 (1997). [23] R. A. Malﬂiet and J. A. Tjon, Nucl. Phys. A127, 161 (1969). [24] A. Stadler, W. Gl¨ockle, and P. U. Sauer, Phys. Rev. C 44, 2319 (1991). [25] W. Saake, Diploma thesis (unpublished), Bochum University, 1992. [26] J. L. Friar, B. F. Gibson, and G. L. Payne, Phys. Rev. C 37, 2869 (1988). [27] A. C. Fonseca and D. R. Lehman, in Proceedings of the 14th International IUPAP Conference on Few-Body Problems in Physics, Williamsburg, VA 1994,Ed. Franz Gross (AIP, New York, 1995).

15

TABLES Paris potential partial wave 1s

0

3s

1

(Eµ , lµ ) 0

100

500

-100

-200

ǫd

(100,0)

(125,2)

(425,2)

(-50,0)

1 p ,3 p 1 1

10

-50

150

300

-150

3p

0

10

-50

150

350

-150

3p

2

(10,1)

(40,3)

(75,1)

(75,3)

(175,1)

10

-50

150

300

-150

− 3 d1

− 3 f2

1 d ,3 d 2 2

(-50,2)

(175,3)

(300,1)

Bonn A and B potentials partial wave 1s

0

3s

1

− 3 d1

1 p ,3 p ,3 p 1 0 1 3p

2

− 3 f2

1 d ,3 d 2 2

(Eµ , lµ ) 0

100

300

-100

-50

ǫd

(50,2)

(100,0)

(300,2)

(-50,0)

10

-50

150

300

-150

(10,1)

(10,3)

(75,1)

(75,3)

(150,1)

10

-50

150

300

-150

(-50,2)

(150,3)

(200,1)

TABLE I. Approximation energies Eµ used in the EST representations of the Paris, Bonn A and Bonn B potentials. ǫd refers to the deuteron binding energy. Eµ are lab energies in MeV. In case of coupled partial waves, the boundary condition chosen for the angular momentum lµ of the initial state (cf. Ref. [4]) is also specified.

16

Channel

Subsystem

l

s

jπ

τ

K

L

1

1s

0

0

0

0+

1

1/2

0

2

3s

1

0

1

1+

0

1/2

0

3

3s

1

0

1

1+

0

3/2

2

4

3d 1

2

1

1+

0

1/2

0

5

3d 1

2

1

1+

0

3/2

2

6

3p

0

1

1

0−

1

1/2

1

7

1p

1

1

0

1−

0

1/2

1

8

1p

1

1

0

1−

0

3/2

1

9

3p

1

1

1

1−

1

1/2

1

10

3p

1

1

1

1−

1

3/2

1

11

1d 2

2

0

2+

1

3/2

2

12

1d 2

2

0

2+

1

5/2

2

13

3d 2

2

1

2+

0

3/2

2

14

3d 2

2

1

2+

0

5/2

2

15

3p

2

1

1

2−

1

3/2

1

16

3p

2

1

1

2−

1

5/2

3

17

3f

2

3

1

2−

1

3/2

1

18

3f

2

3

1

2−

1

5/2

3

TABLE II. Quantum numbers of the three-body channels.

17

# Meshpoints

j ≤ 1+

j≤1

j≤2

6

-8.3318

-8.1383

-9.0551

12

-7.3571

-7.1376

-7.4330

24

-7.3150

-7.0913

-7.3688

36

-7.3156

-7.0919

-7.3688

40

-7.3156

-7.0919

-7.3688

TABLE III. Triton binding energies (in MeV) with the Paris (EST) potential (56555557). The notation (56...) specifies the employed separable representation as explained in Sec. III.

18

j ≤ 1+

j ≤ 2+

j≤1

j≤2

Et (MeV)

P(S)

P(S’)

P(P)

P(D)

Paris (EST) (11)

-7.451

90.63

1.636

0.042

7.692

Paris (EST) (34)

-7.266

89.88

1.652

0.065

8.402

Paris (EST) (56)

-7.316

89.90

1.634

0.064

8.401

Paris (W-matrix)

-7.300

90.22

1.450

0.064

8.265

Paris

-7.297

89.88

1.625

0.066

8.428

Paris-r

-7.310

89.88

1.623

0.066

8.428

Paris (EST) (1111)

-7.464

90.62

1.636

0.042

7.704

Paris (EST) (3444)

-7.375

89.87

1.618

0.066

8.447

Paris (EST) (5644)

-7.424

89.89

1.601

0.066

8.444

Paris (EST) (5655)

-7.426

89.89

1.600

0.066

8.446

Paris (W-matrix)

-7.343

90.21

1.436

0.064

8.288

Paris

-7.408

89.87

1.591

0.068

8.474

Paris (EST) (11111)

-7.464

90.83

1.468

0.044

7.658

Paris (EST) (34333)

-7.074

90.28

1.492

0.064

8.167

Paris (EST) (56444)

-7.093

90.30

1.488

0.062

8.154

Paris (EST) (56555)

-7.092

90.30

1.488

0.062

8.153

Paris (W-matrix)

-7.150

90.51

1.301

0.066

8.123

Paris

-7.103

90.28

1.468

0.063

8.193

Paris (EST) (11111111)

-7.549

90.61

1.459

0.047

7.879

Paris (EST) (56555557)

-7.369

90.14

1.420

0.063

8.379

Paris (W-matrix)

-7.088

90.54

1.366

0.062

8.034

Paris

-7.378

90.11

1.403

0.064

8.418

TABLE IV. Triton wave function components for the Paris potential. The notation (56...) etc. specifies the employed separable (EST) representation as explained in Sec. III. The result for Paris-r is taken from Ref. [26].

19

j ≤ 1+

j ≤ 2+

j≤1

j≤2

Et (MeV)

P(S)

P(S’)

P(P)

P(D)

Bonn A (EST) (11)

-8.350

92.75

1.415

0.028

5.811

Bonn A (EST) (44)

-8.347

92.35

1.427

0.034

6.188

Bonn A (EST) (56)

-8.380

92.31

1.432

0.035

6.220

Bonn A (W-matrix)

-8.371

92.01

1.385

0.041

6.565

Bonn A

-8.378

92.32

1.426

0.035

6.217

Bonn A (EST) (1111)

-8.360

92.74

1.415

0.027

5.820

Bonn A (EST) (4444)

-8.411

92.35

1.411

0.034

6.204

Bonn A (EST) (5644)

-8.444

92.31

1.415

0.034

6.236

Bonn A (W-matrix)

-8.399

92.00

1.380

0.039

6.578

Bonn A

-8.443

92.32

1.411

0.035

6.235

Bonn A (EST) (11111)

-8.298

92.95

1.248

0.030

5.772

Bonn A (EST) (44444)

-8.083

92.72

1.252

0.037

5.995

Bonn A (EST) (56444)

-8.115

92.68

1.254

0.037

6.027

Bonn A (W-matrix)

-8.160

92.36

1.209

0.043

6.391

Bonn A

-8.127

92.69

1.248

0.037

6.029

Bonn A (EST) (11111111)

-8.395

92.81

1.264

0.031

5.895

Bonn A (EST) (56444445)

-8.285

92.59

1.236

0.037

6.135

Bonn A

-8.295

92.59

1.231

0.037

6.138

TABLE V. Triton wave function components for the Bonn A potential. (56...) etc. specifies the employed separable (EST) representation as explained in Sec. III.

20

j ≤ 1+

j ≤ 2+

j≤1

j≤2

Et (MeV)

P(S)

P(S’)

P(P)

P(D)

Bonn B (EST) (11)

-8.209

91.95

1.361

0.035

6.659

Bonn B (EST) (44)

-8.137

91.36

1.369

0.047

7.224

Bonn B (EST) (56)

-8.170

91.34

1.373

0.048

7.236

Bonn B (W-matrix)

-8.161

91.21

1.286

0.053

7.448

Bonn B

-8.165

91.35

1.368

0.049

7.235

Bonn B (EST) (1111)

-8.219

91.94

1.360

0.034

6.668

Bonn B (EST) (4444)

-8.197

91.35

1.354

0.047

7.245

Bonn B (EST) (5644)

-8.230

91.34

1.357

0.048

7.256

Bonn B (W-matrix)

-8.190

91.21

1.286

0.053

7.448

Bonn B

-8.226

91.34

1.354

0.049

7.257

Bonn B (EST) (11111)

-8.159

92.14

1.200

0.037

6.620

Bonn B (EST) (44444)

-7.855

91.75

1.216

0.048

6.985

Bonn B (EST) (56444)

-7.884

91.74

1.218

0.048

6.700

Bonn B (W-matrix)

-7.926

91.56

1.143

0.054

7.247

Bonn B

-7.899

91.74

1.210

0.049

7.001

Bonn B (EST) (11111111)

-8.282

91.96

1.208

0.039

6.791

Bonn B (EST) (56444445)

-8.088

91.61

1.189

0.048

7.149

Bonn B (W-matrix)

-7.919

91.56

1.143

0.052

7.249

Bonn B

-8.103

91.62

1.184

0.048

7.152

TABLE VI. Triton wave function components for the Bonn B potential. (56...) etc. specifies the employed separable (EST) representation as explained in Sec. III.

21

FIGURES

channel 1

channel 2

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

40

0

30

-10

20

-20

10

-30 2.5

2.5

2

0 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2

-40 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 3

channel 4

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0.5

1

0.4

0.75

0.3 0.5 0.2 0.25

0.1 2.5

0 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2.5

0

2

2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 5

channel 6

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.05

0.06

0 -0.05

0.04

-0.1 0.02

-0.15

0

-0.2

2.5

2.5

2 2.5

2

1.5 2

1.5 -1

q [fm ]

1 1

0.5

0.5

2.5

-1

p [fm ]

1.5 2

1.5 -1

0 0

q [fm ]

1 1

0.5

0.5

-1

p [fm ]

0 0

FIG. 1. Graphs of the triton wave function obtained with the Paris (EST) potential. The channels are defined in Table II.

22

channel 7

channel 8

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.5 0.4 0.3 0.2 0.1 2.5

0

2 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2.5 2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 9

channel 10

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.5

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

0.4 0.3 0.2 0.1 2.5 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2.5

0

2

2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 11

channel 12

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0.3 0.3 0.2 0.2 0.1

0.1 2.5

0 2.5

1.5 2

1.5 -1

q [fm ]

1 1

0.5

0.5

2.5

0

2

2.5

-1

p [fm ]

2 1.5 2

1.5 -1

0 0

q [fm ]

FIG. 2. – continued.

23

1 1

0.5

0.5 0 0

-1

p [fm ]

channel 13

channel 14

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0

0.1 0.075

-0.05

0.05 -0.1

0.025 2.5

2.5 0

2

-0.15 2.5

1.5 2

1.5 -1

1 1

q [fm ]

0.5

0.5

2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 15

channel 16

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.15

0 -0.1

0.1

-0.2 -0.3 -0.4

0.05

-0.5

2.5

-0.6

2.5

2 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2

0 2.5

-1

p [fm ]

1.5 2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 17

channel 18

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0

0.01

-0.01

0.005

-0.02 0

-0.03 2.5

-0.04 -0.05 2.5

-0.005

2

-0.01 2.5

1.5 2

1.5 -1

q [fm ]

1 1

0.5

0.5

-1

p [fm ]

2.5 2 1.5 2

1.5 -1

0 0

q [fm ]

FIG. 3. – continued.

24

1 1

0.5

0.5 0 0

-1

p [fm ]

channel 1

channel 2

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

40

0

30

-10

20

-20

10

-30 2.5

2.5

2

0 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2

-40 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 3

channel 4

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0.5

1

0.4

0.75

0.3 0.5 0.2 0.25

0.1 2.5

0 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2.5

0

2

2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 5

channel 6

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.05

0.06

0 -0.05

0.04

-0.1 0.02

-0.15

0

-0.2

2.5

2.5

2 2.5

2

1.5 2

1.5 -1

q [fm ]

1 1

0.5

0.5

2.5

-1

p [fm ]

1.5 2

1.5 -1

0 0

q [fm ]

1 1

0.5

0.5 0 0

FIG. 4. Same as Fig. 1 but for the Bonn A (EST) potential.

25

-1

p [fm ]

channel 7

channel 8

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.5 0.4 0.3 0.2 0.1 2.5

0

2 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2.5 2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 9

channel 10

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.5

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

0.4 0.3 0.2 0.1 2.5 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2.5

0

2

2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 11

channel 12

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0.3 0.3 0.2 0.2 0.1

0.1 2.5

0 2.5

1.5 2

1.5 -1

q [fm ]

1 1

0.5

0.5

2.5

0

2

2.5

-1

p [fm ]

2 1.5 2

1.5 -1

0 0

q [fm ]

FIG. 5. – continued.

26

1 1

0.5

0.5 0 0

-1

p [fm ]

channel 13

channel 14

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0

0.1 0.075

-0.05

0.05 -0.1

0.025 2.5

2.5 0

2

-0.15 2.5

1.5 2

1.5 -1

1 1

q [fm ]

0.5

0.5

2 1.5

2.5

-1

p [fm ]

2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 15

channel 16

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ] 0.15

0 -0.1

0.1

-0.2 -0.3 -0.4

0.05

-0.5

2.5

-0.6

2.5

2 1.5

2.5

2

1.5 -1

1 1

q [fm ]

0.5

0.5

2

0 2.5

-1

p [fm ]

1.5 2

1.5 -1

0 0

1 1

q [fm ]

0.5

0.5

-1

p [fm ]

0 0

channel 17

channel 18

3

3

Ψ(p,q) [fm ]

Ψ(p,q) [fm ]

0

0.01

-0.01

0.005

-0.02 0

-0.03 2.5

-0.04 -0.05 2.5

-0.005

2

-0.01 2.5

1.5 2

1.5 -1

q [fm ]

1 1

0.5

0.5

-1

p [fm ]

2.5 2 1.5 2

1.5 -1

0 0

q [fm ]

FIG. 6. – continued.

27

1 1

0.5

0.5 0 0

-1

p [fm ]