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Resonances in the three-neutron system

A study of 3-body resonances has been performed in the framework of configuration space Faddeev equations. The importance of keeping a sufficient number of terms in the asymptotic expansion of the resonance wave function is pointed out. We investigated three neutrons interacting in selected force components taken from realistic nn forces.


I. INTRODUCTION
Three-nucleon (3N) resonances have not yet been firmly established. The situation up to 1987 has been collected in [1]. A recent investigation [2] of the process 3 He(π − , π + )3n found no evidence of a resonance state of three neutrons. Earlier investigations [3], [4] pointed to resonances in the three neutron (3n) and three proton (3p) systems. The double charge exchange process on 3 He was also investigated in [5] criticizing previous work but pointing again to a 3n resonance, now around (20 − i 20) MeV. For the nnp system an excited state around 7 MeV with a width of Γ = 0.6 ± 0.3 MeV has been suggested in [6]. A variational study based on complex scaling and simplified nucleon-nucleon (NN) forces was carried through in [7] with the prediction of a 3n resonance at E = (14 − i 13) MeV for a J π = 3/2 + state. Earlier theoretical studies [8], [9] might be useful in various aspects, but are not conclusive with respect to the actual position of 3n resonances. A virtual state of 3 H with the quantum numbers of the ground state has been investigated in [10] by analytically continuing the momentum space Faddeev equation and even estimating three-nucleon force (3NF) effects. In the framework of the hyperspherical harmonic method and using complex scaling with model potentials a subthreshold 3n resonance for J π = 1/2 − has been located in [11]. Recently mathematical foundations have been laid on the analytical continuation of the three-body Faddeev equations into unphysical energy sheets [12] and an application thereof to the 4 He trimer appeared in [13].
It is a theoretical challenge to determine the locations of the lowest lying 3N resonances on unphysical sheets based on the most modern nuclear forces. Those forces are very successful in describing 3N scattering observables [14] which leads to the expectation that the corresponding 3N Hamiltonian will also locate 3N resonances with a high degree of reliability. Like for the 3N bound state energy, where some theoretical underbinding results and 3NF's are used to bridge that gap to the data point, also 3N resonance positions might finally be fine-tuned in this manner. These future rigorous results would also be an important guidance for experiments and would clarify the hitherto controversial situation. For the 3N system the bound state has been studied extensively in the last decades resulting in numerically precise and converged solutions of the 3N Faddeev equations [15], [16], [17], [18], [19]. Energy eigenvalues and wave functions have been generated with most modern nuclear Hamiltonians in momentum and coordinate space. Also numerically exact solutions of 3N Faddeev equations for continuum states with the same Hamiltonians are now available in momentum space for a wide range of energies [14]. The first model solutions also in coordinate space appeared [20] and variational approaches matured and provide equally accurate solutions [21].
3N resonances, however, have not yet been investigated in a comparable rigorous manner and based on realistic NN forces to the best of our knowledge. In this article we would like to take first steps into such a direction.
In the two-nucleon (2N) system the exact form of the wave function in configuration space is known outside the range of the potential. In the 3N system only asymptotic expansions are known [22], [23] and methods how to apply them with sufficient accuracy, for instance in Faddeev equations, have still to be elaborated in that framework. There are first very promising results for 3N scattering above the 3N breakup threshold [20] though up to now only for simple S-wave model forces. To the best of our knowledge 3N resonance wave functions, which asymptotically oscillate and grow exponentially, have not yet been determined as solutions of Faddeev equations and realistic forces. First attempts in this direction have been done in [24], however with very simple forces.
In this study we investigate 3-body resonances as solutions of configuration space Faddeev equations. The largest obstacle to find the resonance positions with sufficient precision is the approximate form of the resonance boundary condition. Ab initio it is not known at which coordinate values this form is valid with sufficient accuracy. For resonances this problem can be overcome in principle using the complex scaling method (CSM) [25], [26]. Thereby all relative coordinates are "rotated" into the complex plane leading to bound-state like, exponentially decreasing resonance wave function. This "rotation" of the coordinates into the complex plane works well for problems in atomic physics with potentials of inverse power laws. It will be shown in this article that in nuclear physics a straighforward application of CSM in the configuration space Faddeev form leads to serious numerical problems when the usual realistic NN interactions of Yukawa type and hard core behavior are used. Whether feasible modifications thereof can be found remains to be seen. Right now our experience is such, that the resonance energies can be gained with sufficient precision only if one does not leave the real r-axis and if one guaranties the exact boundary condition as close as possible by using a sufficient number of terms in its asymptotic form.
In section II we explain in detail the problems encountered when one applies the CSM to the nuclear resonances using the 2N system as a play ground. The exact knowledge of the boundary conditions for this system allows one to check assumptions which can later be applied to 3N resonances. The 3N equations and the method for their solution are presented in section III. There we discuss results for 3 bosons interacting with simple, S-wave forces and more importantly for 3n's interacting via realistic neutron-neutron (nn) interactions. A summary is given in section IV. Information on the technical performance can be found in the Appendices A and B.

II. RESONANCES IN THE 2N SYSTEM
Searching resonances in a system of two-nucleons is a well defined problem and numerically well under control [27]. In spite of that we discuss it here and present some results due to its pedagogical simplicity, which enables us to present and test approaches which might be useful for finding resonances for three or more nucleons. In addition we can introduce our notation.
The 2N resonances are associated with the poles of the S-matrix which are embedded in the fourth-quadrant of the complex k-plane (Re(k)> 0, Im(k)< 0). They are solutions of the time-independent Schrödinger equation without incoming wave and the outgoing wave increasing exponentially at infinity. We introduce the coordinate space partial wave basis |r, α 2 >≡ |r, (ls)jt > where and where the orbital angular momentum l and total spin s are coupled to the total angular momentum j. The isospin of the two nucleons is denoted by t. Then the resonance wave function ψ( − → r ) to a given j and t has the form It is written in terms of the reduced wave functions φ α 2 (r) which fulfill the radial Schrödinger The resonance conditions are such that ψ has to be regular at the origin and purely outgoing. For the reduced amplitudes this amounts to and outside the range of the interaction. Here h (1) l (kr) is the spherical Hankel function. As we shall demonstrate below it is mandatory to either use the exact form of the boundary condition (5) or to take into account at least the leading orders to find the resonances in the 2N system with sufficient accuracy. In Eq.(3) m is the nucleon mass and k 2 ≡ m h 2 E. The complex energy E is embedded in the lower half plane of the second energy sheet. The boundary conditions (4) and (5) apply also to bound states with real, negative eigenvalues E. The matrix elements of the NN interaction are defined in terms of V α 2 α ′ 2 (r) by A method of solving Eq. (3) is displayed in [28]. One chooses suitable grid points in r and interpolates φ α 2 (r) in terms of cubic splines. The boundary conditions (4) and (5) can be conveniently incorporated either by the absence of certain expansion coefficients in the spline expansion or by suitable relations among them. Then choosing appropriate collocation points in r one ends up with a homogeneous algebraic set of equations. They can be handled by standard methods [28].
In systems with more than 2 nucleons the boundary conditions are known only via an asymptotic expansion and one has therefore to work with approximate boundary conditions.
The knowledge of the exact resonance solutions in the 2N system allow now to study the question pivotal for 3N systems: how precisely does one need to know the boundary conditions in order to get the resonance energies with sufficient accuracy? To study this problem one can use the series representation for the spherical Hankel functions and write the resonance wave function outside the interaction region in a form which resembles the asymptotic boundary condition in the 3N system (see Section III, Eq. (21)) with a m being unknown constants independent from k and r. Here, contrary to the 3N system, the number of terms in the series is restricted by the orbital angular momenta contributing to the given total angular momentum j of a resonance.
In Table I we present 2N resonance positions in different NN states where the exact boundary condition have been imposed at different matching radii outside the interaction region. The NN interaction Reid93 [29] has been used and has been multiplied with a factor λ V (shown in column 2 of Table I For resonances the problem of a possibly approximate nature of the boundary condition can be overcome in principle using the complex scaling method (CSM) [25], [26]. This will now be illustrated for 2 nucleons because of its simplicity. In the 2N system one replaces the relative coordinate r by For real positive angles α the asymptotic form e ikρ with Im(k) < 0 can be changed to an exponentially decreasing function. Therefore a resonance wave function analytically continued to those complex coordinates can be treated like a bound state with its well known boundary condition. Then it is probably enough to restrict the boundary condition (8) to the first term m = 0. This "rotation" of the coordinate into the complex plane is sufficient for problems in atomic physics with Coulomb potentials. In nuclear physics the potentials decrease exponentially and the transformation (9) turns then e −µ r into e −µ(cos(α)+isin(α))r .
Therefore for α > 0 the range of the potential will increase, which is counterproductive.
We firstly illustrate the power of the ECSM in the 2N system using the Reid 93 NN potential in the state 3 P 0 . As we have seen in Table I Tables I and II). The result is improved and we can gain the correct result with the required precision using CSM with Im(α) = 0 (see Table II). Note that in Table II we stick to the asymptotic form (m = 0) in view of the fact that for 3 bodies we also have only approximate forms. If we now allow for negative Im(α)'s then as can be seen in Table II we gain the correct eigenvalue even choosing a very small matching radius.
We also see from Table II [29] or Reid soft core potential [30]. The analytical continuation to complex r-values leads to an oscillating behavior of the potential, which drastically increase with increasing Re(α), with unstable numerics as a consequence. We examplify this behavior in Fig. 1 for the Reid93 potential.
We checked several of the existing NN potentials and found always this problem ( potentials according to Gogny [31], R.de Tourreil and D.W.L. Sprung [32], H. Eikemeier and H.H. Hackenbroich [33], and R. de Tourreil [34]). In case of the AV14 [35] and AV18 [36] potentials the situation is even worse due to a singularity for complex r-values in their parametrization of the hard core part. Also Gauss-parametrizations are not an exception. In this case the form exp(−ar 2 ) limits the complex scaling angle Re(α) to Re(α) < 45 o .
One possibility to generalize the CSM in order to avoid the above problems is the so called "smooth-exterior" complex scaling (SECS) [26]. The idea is to perform the complex rotation only outside the region of interaction. In this way the analytical continuation of the potential into the complex coordinate plane is avoided. In practice this can be achieved for instance by performing the following transformation (note the printing errors in [26] leading to false expressions (2.1.30) and (2.1.32)) with This transformation leads to the smooth-exterior-scaled Hamiltonian and The case λ = 0 corresponds to usual CSM r ′ = r e iα [26]. Taking r 0 outside the region of the NN interaction and choosing a proper value for λ a smooth-exterior-scaling path results, which avoids the oscillating changes of the NN potential. We present in Table III the effectiveness of this approach by aplying it to the 1 D 2 resonance state for the Reid93 NN potential with λ V = 5.0. It is enough to keep only the first term in the boundary condition (8) to reproduce the correct resonance position when a sufficiently large value of the scaling angle α is chosen. Now, however, contrary to CSM, no numerical problems arise when large values of α are used. In spite of its success in the 2N system we think that the SECS cannot be easily applied in 3N systems. We could not overcome problems arising from the permutation operator (see Section III).

III. THREE-BODY RESONANCES
Here H 0 is the kinetic energy, V the two-body force, ψ a Faddeev component and P the sum of two permutation operators such that the total, properly symmetrized wave function it is convenient to introduce polar coordinates x ≡ ρcosθ and y ≡ √ 3 2 ρsinθ. If we express the Faddeev component ψ( − → x , − → y ) in terms of partial wave amplitudes |α > which specify angular momenta and isospin quantum numbers, then we obtain and Eq.(16) turns into a coupled set of partial-integrodifferential equations [28] ( and The coupling results from the permutation operator P and from possible spin dependencies in V. The evaluation of the permutation operator matrix elements < xyα|P |x ′ y ′ α ′ > and the resulting expression for the kernel K βγ (θ, θ ′ ) is presented in Appendix A.
Similarily as in the 2-body case the resonance conditions are such that ψ(x, y) ≡ φ(x,y) xy has to be regular at the origin and purely outgoing in all channels. For the reduced amplitudes φ(ρ, θ) this amounts to [28,38] and for large distances The first term in Eq. (21) occurs only in channels α, where a two-body bound state u(x) exists. The second part describes the behavior in the 3-body breakup channel. The corre- where m is the nucleon mass and E 2 the two-body binding energy. The boundary conditions (21) apply also to bound states with real eigenvalues E which are located below all two-body For resonances E is complex and the eigenvalues E close to the real axis of the physical sheet are of interest. They either lie in second sheets which are accessible to the right of E 2 -values (two-body fragmentation cuts) or in the lower half plane of a second sheet, which is accessible through the 3N-breakup cut to the right of E = 0. For a recent description of the cut-structure see for instance [24].
An efficient method for solving Eq. (18) is very well displayed in [28]. One chooses suitable grid points in ρ and θ and interpolates φ(ρ, θ) in terms of cubic splines. Similarily as in the 2-body case the boundary conditions (20) and (21) can be conveniently incorporated either by the absence of certain expansion coefficients in the spline expansion or by suitable relations among the coefficients. Then choosing appropriate collocation points in ρ and θ one ends up in a homogeneous algebraic set of equations. For realistic NN forces the dimension of the matrix is very large and the tensor product method [39] is a key-technique. We use a Lanczos type iteration method [40] and of course take advantage of the sparceness property of the matrices. It is advantegeous to work with further reduced amplitudes F defined as φ(ρ, θ) ≡ ρe ikρ F (ρ, θ), if one treats resonances below the cut with Re E > 0. In Appendix B we present the details of the above procedure.
The main problem in solving Eq. (18) is the approximate nature of the asymptotic form (21). Ab initio it is not known at which ρ-value this form is valid with sufficient accuracy. For a recent discussion see [38]. Also it is a priori not known how many terms should be included in the asymptotic expansion (21) for a particular 3N resonance state. One suspects of course that including more terms one approaches more closely the exact boundary condition. The importance of the terms in Eq.(21) going with increasing powers of 1 kρ depends not only on the resonance energy and the matching value for ρ but also on the magnitudes of the a priori unknown functions A(θ), B(θ), · · ·. For resonances commonly these problems are expected to be overcome using the complex scaling method (CSM) [25]. However, due to the restrictions imposed on this method by the behavior of the NN forces as discussed in Section II we found that this method can be applied in the framework of the configuration space Faddeev equations only for rather narrow resonances. Thereby all relative coordinates ( x and y in the 3-body case) are rotated into the complex plane. This amounts to ρ → ρ ′ = ρe iα where α is restricted to rather small values. An example for realistic nn forces is mentioned at the end of this Section. However, if the forces are very simple, CSM can work as will be illustrated below.
One is intended to hope that SECSM which proved to be very efficient in the 2-body case will help here, too. However, according to our insights up to now this approach leads to problems we could not solve when calculating the permutation operator P and when performing the transition from cartesian to polar relative coordinates.
The only possibility left is to look at solutions of Eq.(18) without leaving the real r-axis and taking the boundary condition (21) with a sufficient number of terms into account in order to reach as close as possible the exact boundary condition. In this manner one can hope to get convergence for the resulting resonance position.
We checked our configuration space code by calculating the energy of the bound state of 3 H nuclei with the Reid 93 NN potential taking different NN force components into account. In Table IV we present our numbers together with results of momentum space Faddeev calculations [41]. The very good agreement between both approaches is clearly seen.
In the first step we considered 3 bosons interacting via Gaussian S-wave forces They are such that a barrier is built in, which is an obvious and trivial mechanism generating a resonance. That interaction does not support a two-body bound state and in the asymptotic form (21) the first term is absent. For 3 bosons in the J π = 0 + state and interacting by S-wave forces only the kernel K αβ = 1 4π . This model has been used in [42] in applying the stochastic variational method. The resonance energy found there is E = (8.60 − i 1.84) MeV. In this case CSM works. For instance for a matching radius of ρ = 10 fm and taking the A-and B-terms in (21) into account we achieved the results shown in Table V. For Re(α) = 40.0 o we are close to the result found in [42] but not close enough.
A smaller matching radius, however, like ρ = 5 fm does not work. The correct result can however be achieved if in addition we turn on Im(α) < 0. This is demonstrated in Table V.
We illustrate in Figs.2 and 3 typical resonance amplitudes F working with real coordinates or with complex coordinates. In both cases the matching radius has been chosen as 5 fm. In the first case this is not sufficient to achieve the correct complex energy eigenvalue, whereas in the second case the amplitude is already sufficiently damped at ρ = 5 fm and the correct energy eigenvalue is accessible at such a small matching radius.
In this simple case one can study the problem how many terms A, B, · · · in the boundary condition (21) should be taken into account to get the correct resonance position. Of course it will depend on the value of the matching radius ρ. We present in Table VI resonance positions obtained with increasing number of terms in the boundary condition (21) and different matching radii. For the details to include arbitrary number of terms in the asymptotic condition we refer to Appendix B. It is seen from Table VI that for matching radii > 10 fm it is practically sufficient to take only the two leading terms going with A and B into account in order to get the correct resonance position. We also see that the restriction to the A term alone is totally insufficient. Unfortunately this example is far from reality in nuclear physics.
It does not contain a short range repulsion and the mechanism for creating a resonance in few nucleon systems is different in reality, where it is not caused by a barrier in the potential itself.
Now we investigate the case of three interacting neutrons in states with different total angular momenta and parity. We allowed the nn force to act in the states 1 S 0 , 3 P 0 and 3 P 2 − 3 F 2 . They are of attractive nature while the repulsive ones were neglected. This is only a feasibility study and should provide a first orientation. In the first step the nn forces were chosen as a smooth, local NN force of ref. [31], called in the following Gogny potential. It is a rather simple approach to NN phase-shifts through forces of central, tensor and spin-orbit type supplemented by a L 12 term, all of which of Gaussian form. To initiate the search process for the position of the resonance energy we artificially enhanced the force components in the three states such that all of them support a two-neutron bound state at the same energy. An example is shown in Table VII for a J π = 3/2 − state of three neutrons. For these enhancement factors also the 3n system is bound. Then we reduce the enhancement factors (some intermediate cases are shown in Table VII), which traces out a bound state trajectory. In this case we see that the 3n bound state still exists even when all 2n subsystem bound states have disappeared. Then we can neglect the first term in the boundary condition (21). Decreasing further the enhancement factor(s) (in this specific case only for 3 P 2 − 3 F 2 ) the energy eigenvalue goes into the second sheet and traces out a resonance trajectory. The search for the resonance position corresponding to the actual nn force is finished when the enhancement factors in all nn force components have reached the values λ i = 1.0. As we shall see in the case of three neutrons we face numerical instabilities before all λ ′ s reach the value 1. We can present the resonance trajectory only partially up to the point, where we encounter those instabilities. We expect that they are probably caused by the difficulty to fulfill the resonance condition, which requires that the exponentially decreasing, purely incoming wave is much smaller than the exponentially increasing purely outgoing wave. As we see from Table VII the J π = 3/2 − 3n bound state disappears when the enhancement factors arrive at (λ1 S 0 , λ3 P 0 , λ3 P 2 − 3 F 2 ) = (1.0, 1.0, 4.35). In Table VIII the trajectory of the 3/2 − 3n resonance in the second energy sheet is mapped out by decreasing the enhancement factor λ3 P 2 − 3 F 2 in step of 0.05 starting from λ3 P 2 − 3 F 2 = 4.3. The final reliable position reached is for λ3 P 2 − 3 F 2 = 3.6 with the value E 3/2 − 3n = (5.30 − i4.67) MeV.
As can be seen this stable position is only obtained when at least 4 terms in the boundary condition (21) are taken into account. We checked that our particular distribution of ρ and θ points as well as the choice of the matching radius in ρ does not influence the resulting trajectory. The values given in Table VIII are based on a matching radius ρ = 30.0 fm and typically N ρ = 30 and N θ = 25 points distributed by scaling factors S ρ = 1.2 and S θ = 0.9 (see ref. [28]). It is unfortunate that right now we are not able to reach the final position with λ3 P 2 − 3 F 2 = 1. However very likely the width will be quite large. Therefore one cannot expect to see a visible enhancement around some 3n c.m. energy in a reaction producing three neutrons in the state 3/2 − . Similarily to 3/2 − behave the 1/2 − and 3/2 + states based on the Gogny potential.  Table IX. For the 1/2 − and 3/2 + states, similarily as for 3/2 − , the final, converged position is obtained only when at least 4 terms in the boundary condition are taken into account. In Fig.4 we show the trajectories for the three states 1/2 − , 3/2 − , and 3/2 + for the Gogny potential. We note that the trajectories are not unique and depend how the enhancement factors are changed. Our startegy was to bring all of them together to one as close as possible.
In this article we did not investigate the state 1/2 + , since we found, like in [9] that the 3n bound state disappears earlier than the 2n bound states. In other words, the 3n bound state trajectory will enter into the 2n-n two-body fragmentation cut. We left that study with the additional asymptotic term in Eq.(21) to a future investigation.
It is seen from Table IX that the 3/2 + state for the Gogny potential is relatively narrow.  Table X). Increasing it, however, the resulting energy eigenvalue is totally wrong what can be traced back to the increasing oscillations of the NN potential resulting from its analytical continuation to the complex r-values.
In the last step we checked how the results depend on the NN interaction used. We choose a modern, realistic NN interaction as given by the Reid Table XI. In spite of the fact that the particular values of the λ V 's change of course, the resulting picture is basically the same as for the Gogny potential. We are still too far away from all λ i = 1 in order to definitely say that all the widths will be quite large. But this will likely be the case. Again to achieve converged results one has to take at least 4 terms in the boundary condition (21) into account.

IV. SUMMARY
We investigated the configuration space Faddeev equations with the aim to determine the location of 3-neutron resonances. Realistic nn forces have been used. The asymptotic form in the 3n breakup channel is known only in the sense of an asymptotic expansion. We found that several terms thereof have to be included in order to get stable and reliable results, which are independent of the matching radius. The search for the resonance positions was carried through by artificially enhancing 2n force components such that a 3n bound state exists.
Then by reducing those enhancement factors we followed the paths of the energy eigenvalue into the second sheet adjacent to Re(E) ≥ 0. Unfortunately it turned out that numerical instabilities set in before we reached the actual strengths factors 1 for all the individually enhanced force components. Therefore we could not reach the very final positions of the studied 3n resonances in the states J π = 1/2 − , 3/2 − , and 3/2 + . The detailed reason for those instabilities remains to be found out. Despite of that our results already indicate that the imaginary parts of the complex resonance energies will be quite large, so that one should not expect to see resonances in experiments.
We also applied the CSM, which works beautifully in atomic phyics. The complex coordinates turn the exponentially increasing resonance wave function into an exponentially decreasing one, like for a bound state, and the problem with the approximate boundary conditions is avoided. However, for realistic NN interactions the complex scaling transformation r → r e iα increases the interaction range by a factor 1 cosα and leads for scaling angles α greater than about 40 o to a strong oscillations of the potential. This restricts the application of the complex scaling method in more than 2-nucleon systems to rather narrow resonances. In the 2-nucleon system one can avoid the above restriction by performing the complex rotation outside the region of non-zero interaction. Due to the neccessary particle permutation in the 3-body system we were not able to use that idea in the 3-body context. This project deserves further investigations to locate all lowest resonances in the three nucleon systems based on modern NN and possibly three-nucleon forces.

APPENDIX A: PERMUTATION OPERATOR
To evaluate the term on the right hand side of Eq.(16) one needs where the permutation operator is P ≡ P 12 P 23 + P 13 P 23 and the set of discrete quantum numbers abbreviated by β or γ is {(ls)j(λ1/2)I(jI)JM; (t1/2)T M T }. Here l,s, and j are the orbital angular momentum, total spin and total angular momentum in the two-body subsystem 2-3, λ, 1/2, and I are the corresponding quantities for particle 1, and JM denote the conserved 3N total angular momentum and its magnetic quantum number. In isospin space, t refers to particles 2 and 3 and couples with the isospin 1/2 of particle 1 to the total isospin T and its magnetic quantum number M T . Following [27] one obtains for the matrix element of the permutation operator The purely geometrical quantityg r 1 s 1 k αα ′ is given bỹ The superscript cart stands for cartesian andl ≡ 2l + 1. To arrive at the kernel K βγ on the right-hand side of Eq.(18) one has to evaluate We introduce polar coordinatesx and changing the u to the θ ′ integration one arrives at From Eq.(A9) follows that
Due to the properties of the cubic splines as given in [28] one has The boundary conditions at the origin (20) imply that a α,0,n = 0 Writing the asymptotic expansion (21) for F α (ρ, θ) under the assumption that no two-body bound state exists in any channel α results in when N t terms are included in the asymptotic form.
Imposing the boundary condition (B5) at ρ = ρ max implies the dependence of a α,m=M +1,n on a α,m=M,n and B α l (θ). In order to remove the unknown B α l (θ) functions the boundary condition has to be imposed on the last N t points of the ρ-grid. This leads to the dependence of a α,m=M +1,n on a α,m=M,n , · · ·, a α,m=M −2(Nt−1),n . In the following the form of this dependence is derived.
The asymtotic form for the partial derivative ∂Fα ∂ρ is according to (B5) and for l ≥ 2Ã Taking the partial derivative of (B2) with respect to ρ and the form (B6) yields for the last where for n = 2kB α l = B α l (θ k ) and for n = 2k + 1B α l = dB α l dθ | θ=θ k . The set (B10) are N t equations for N t −1 unknownsB α l (θ k ). They allow to write a α,M +1,n in terms of the a α,M −i+1,n , i = 1, · · · , 2(N t − 1) + 1 as where the C i 's are functions of all theÃ l (ρ I−i+1 )'s resulting from (B10).
In this way, incorporating the boundary conditions (B4) and (B5) one can write F α (ρ, θ) on the chosen grids as where, due to the last equation in (B4) we renumbered n = N + 1 by n = N.
Inserting the expansion (B12) with the help of (B1) into Eq. (18) and choosing collocation points ρ p , p = 1, · · · , 2I and θ q , q = 1, · · · , 2K [28] yields with and The number of channels is denoted by N c . This eigenvalue problem (B13) can be treated in a standard way by changing it into the form and searching for that energy at which λ = 1.
To solve Eq.(B16) we used a Lanczos type iteration method [40]. For the application of B and A −1 on a given vector a we applied the tensor product method [39]. Hereby A is decomposed as Then we use the identitȳ and Π, Ξ diagonal.
It follows and approximate (columns 5-7) boundary conditions applied at different matching radii r max . The Reid93 NN potential multiplied with the factors λ V has been used.
r-matching r-matching