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The Faddeev-Yakubovsky symphony
Rimantas Lazauskas* · Jaume Carbonell
Received: date / Accepted: date
Abstract
We briefly summarize the main steps leading to the Faddeev-Yakubovskyequations in configuration space for N=3, 4 and 5 interacting particles.
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The Faddeev-Yakubovsky (FY) equations constitute a rigorous mathematicalformalism for solving the non relativistic N-body problem of Quantum Me-chanics. They were motivated by the drawbacks of the fundamental Schr¨odingerequation in determining the physical solutions, mainly those related with thescattering states, of a 3-body problem. FY equations provide a framework en-abling us to implement additional constrains to the solutions of the Schr¨odingerequation, which guarantee their physical meaning.FY equations should in principle allow us to obtain the exact solutionsof this fundamental equation for an arbitrary number of interacting particleswith the only input of the interparticle potential. The term ”exact” must behere understood in the numerical sense, i.e. allowing numerical methods thatcould provide accurate enough solutions in a controlled converging scheme.Only when such a possibility is ensured one would be able to disentangle theconsequences of a deficient interaction from the consequences of an approxi-mate solution, as was the main aim in the historically first physical problemat the origin of these theoretical developments: the three-nucleon problem.
Rimantas LazauskasIPHC, IN2P3-CNRS/Universit´e Louis Pasteur BP 28, F-67037 Strasbourg Cedex 2, FranceE-mail: [email protected] CarbonellInstitut de Physique Nucl´eaire, Univ Paris-Sud, IN2P3-CNRS, 91406 Orsay Cedex, FranceE-mail: [email protected] Rimantas Lazauskas*, Jaume Carbonell
In practice, and more than 50 years after the publication of Faddeev’s sem-inal work [1,2,3,4,5], the number of interacting particles for which the FYequations have been properly solved remains severely limited to N ≤ The problem faced by the Schr¨odinger’s equation to describe N ≥ V ij ( E − H ) ψ = V ψ H = − N X i =1 ¯ h m i ∆ r i V = N X i In order to edge these problems, L.D. Faddeev derived in 1960s a set of equa-tions, equivalent to (1), which constitutes a rigorous mathematical frameworkfor describing the variety of physical situations involved in N=3. This semi-nal result was later used by S.P. Merkuriev to derive the boundary conditionsin configuration space [9,10] what allowed the first solutions in the case of3-nucleon problem with realistic interactions [11,12,13].A preliminary step consists in isolating the intrinsic dynamics of the three-body Hamiltonian (1). This is achieved by introducing the Jacobi coordinates x α = r µ β,γ m ( r β − r γ ); α = 1 , , . y α = r µ βγ,α m ( r α − R βγ ) , (9)where ( αβγ ) is a circular permutation of (123), µ s,t denotes the reduced massof the system formed by particle clusters s and t , R s the center of mass ofcluster s . An arbitrary mass m is introduced to fix the length scale. They aresupplemented with the center of mass coordinate M R = m r + m r + m r ; M = m + m + m . (10)In terms of them, the solution of (1), ψ factorizes into an intrinsic part Ψ anda plane wave for the center of mass motion ψ ( r , r , r ) = Ψ ( x i , y i ) e i P · R . (11)The intrinsic wave function Ψ , the only one in which we will be interestedhereafter, is a solution of[ E − H ] Ψ ( x i , y i ) = V Ψ ( x i , y i ) , (12)with (in ¯ h /m units) H = − ( ∆ x i + ∆ y i ) . (13)This trivial geometrical operation not only reduces the dimensionality of theproblem, from on R to R , but is the only way to properly disentangle theintrinsic 3-particle energy the from its center of mass excitations. Notice thatthere are 3 different sets of Jacobi coordinates (9) and that the form of intrinsicfree Hamiltonian (13) is independent of a particular choice. They are relatedeach other by orthogonal transformations x α = + c αβ x β + s αβ y β , y α = − s αβ x β + c αβ y β , (14)with c αβ = − r µ α µ β (1 − µ β )(1 − µ α ) , s αβ = ǫ αβ q − c αβ , (15) he Faddeev-Yakubovsky symphony 5 and ǫ αβ = ( − β − α sig ( β − α ).The central idea of Faddeev was to split the wave function in a sum ofthree terms, the so-called Faddeev components (FC), Ψ = N X i More interestingly, and this was the main Faddeev’s result, the T-matrixperturbative expansion (8) can be reordered in such a way that all delta func-tions disappear leading to compact equations.To solve in practice equations (18) one can perform a partial wave expan-sion of each FC in its proper set of Jacobi coordinates Φ i ( x i , y i ) = X α i ϕ α i ( x i , y i ) x i y i Y α i (ˆ x i , ˆ y i ) , (21)where Y α i denote the bipolar spherical harmonics Y α i (ˆ x i , ˆ y i ) = X m x m y < l x m x ; l y m y | l x l x ; LM > Y l x m x (ˆ x i ) Y l y m y (ˆ y i ) , and α i = { l x l y LM } the set of intermediate quantum numbers in the angu-lar (and eventually spin and isospin) couplings. After inserting (21) in (18)and projecting the angular part, one is led with a system of two-dimensionalcoupled integrodifferential equations having the the form (cid:2) q + ∂ x + ∂ y − v eα ( x, y ) (cid:3) ϕ α ( x, y ) = v α ( x ) X β Z +1 − du h (3) αβ ( x, y, u ) ϕ β [ x β ( x, y, u ) , y β ( x, y, u )] , (22)with the effective potential v eα ( x, y ) = v α ( x ) + l x α ( l x α + 1) x + l y α ( l y α + 1) y . The integral kernels h (3) αβ constitutes the key ingredient of the calculation. Theirprecise expressions can be found in [14,30].Faddeev equations have a very simple form for 3 identical particles in theso called S-wave approximation, i.e. where all the angular momenta are set tozero:( q + ∂ x + ∂ y − v α ) φ α ( x, y ) = v α n a X β =1 c αβ Z +1 − du xyx ′ y ′ φ β ( x ′ , y ′ ) , (23)with α, β = 1 , . . . , n a ,2 x ′ ( x, y, u ) = x + 3 y − √ xyu, y ′ ( x, y, u ) = 3 x + y + 2 √ xyu, and c αβ are numerical coefficients which take the values: n a =1 and c =2 for a3 boson system with total angular momentum L=0, n a =1 and c =-1 for a 3fermion system with total spin J=3/2, n a =2 and c = c =1/2, c = c =3/2for a 3 fermion system with total spin J=1/2.Before closing this section, several remarks are in order: he Faddeev-Yakubovsky symphony 7 1. We have conscientiously ignored until now, the existence of three-bodyterms in the potential. They play an important role in the descriptionof nuclear systems but are not relevant in our discussion. The interestedreader can found a sound presentation in [14].2. One of the essential properties of the Faddeev equations in view of fixingthe boundary condition their decoupling in the asymptotic regions of theconfiguration space. This is ensured under the assumption of short rangepairwise interactions. However in presence of Coulomb forces this decou-pling is not guaranteed, at least in the above presented original form, andthe whole formalism could seem questionable. This problem was solvedby Merkuriev [15,16] by introducing an artificial splitting of the Coulombpotential into a short and long range parts by means of a smooth cut-offfunction. The long range part is kept in the left hand side of the Faddeevequations to define the Coulomb asymptotes and the short range partsappear in the right hand side ensuring the decoupling of the equations.Merkuriev approach has proven to be very successful in several nuclearphysics problems like p-d reactions [17] as well as handling purely atomicproblems [18,19,20].3. In parallel with Faddeev work, P.H. Noyes and collaborators had indepen-dently proposed, and properly solved for the bound as well as for elasticscattering cases, the differential form of the same equations [21]. They aresometimes denoted as Faddeev-Noyes equations, specially in their simpli-fied S-wave form (23).4. It is worth noticing that, even for the three-body case, there remain manyunresolved problems, for instance those related with the presence of aninfinite number of the open asymptotic channels. In particular, the verychallenging problem of atomic anti-hydrogen production slow antiprotoncollisions with hydrogen atoms due to the presence of large number of theopen asymptotic channels.The Faddeev solution of the quantum mechanical 3-body problem was ofparamount importance in theoretical physics and definitely set the foundationsfor an ab initio solution of the many particle systems. 21 3 Fig. 1 Tree diagram representing the only possible way to build a 3-body cluster endingwith an interacting pair (12) Nevertheless it remains a very particular case which somehow hides thecomplexity of the N-body problem as well as the road for its generalization Rimantas Lazauskas*, Jaume Carbonell to larger N. In the Faddeev equations, 3 is the number of particles, the num-ber of interacting pairs, the number of Jacobi sets, as well as the numberof ways for breaking (or building!) a 3 body cluster (123) into (or from) alower rank sub-clusters [22]. This later characteristic, of particular relevancein the N-body case, can indeed occur according to the 3 different ”partitions”(12)3,(13)2,(23)1 which have in fact the same topological properties, i.e. thatcan be related to each other by permutation operators. It is convenient torepresent these different topological types in form of ”trees” diagrams, whichin the N=3 case reduces to the single one represented in Fig. 1.Thus, in this case, the number of particles encodes in fact very differentproperties and make difficult to disentangle the different role they play in thegeneral theory. This is however not the case in general as it will be illustratedin the following sections. Few years after Faddeev’s pioneer work, Yakubovsky [5] proposed a consistentscheme to build a set of equations which should allow the solution of the generalN-body problem. His demonstration was based on induction with respect tothe number of particles and this requires the previous solution of the N-1, N-2, . . . , 2 problems. After several unfruitful attempts [23], Yakubovsky’s resultwas a real tour de force to get rid of the δ -like singularities in the perturbativeexpansion of the N-body T-matrix, which disappear after N-2 iterations of theproposed equations.Rather than reproducing this general approach, which remain quite ab-struse even after some pedagogical efforts [24], we will detail in this section –without any formal demonstration – the main steps in deriving the differentialequations for the N=4 case.The starting point is the intrinsic Schr¨odinger’s equation for N=4:( E − H ) Ψ = V Ψ ; V = X i Upper part: Tree diagram representing the two possible topological ways for breakingfully interacting 4-body cluster (1234) into substructures. Lower part: Jacobi coordinatescorresponding to each cluster partition. By permuting particle indexes, one obtains 6x2=12 K-like components and3x2=6 H-like, which give the 18 FY coupled equations (32). In case of iden-tical particles permutation symmetry allows to express all 12 K-like compo-nents from any of them (idem for H-like components); thus one may reduce theproblem to two coupled equations for the two FYC of fig Fig. 2. The Jacobi co-ordinates, associated with each FY component ( x K , y K , z K ) or ( x H , x H , x H ), Usually denoted by K lij,k ≡ Φ lij,k and H ij,kl ≡ Φ ij,kl he Faddeev-Yakubovsky symphony 11 are schematically represented in the lower panel of Fig. 2. They are denotedgenerically by q and have the general form given in (9), i.e. q s,t = r µ st m ( R s − R t ) , (34)where R i denotes the center of mass positions of the two clusters (eventuallysingle particles) connected by each coordinate and µ st their reduced mass.A practical solution of the FY equations can be achieved by means of apartial wave expansion of the FY components in analogy with the 3-body case(21). One ends with a system of tridimensional integrodifferential equations ofthe form (cid:0) q + ∆ x,y,z − v eα (cid:1) ϕ α ( x, y, z ) = v α X β Z +1 − dudv h (4) αβ ( x, y, z, u, v ) ,ϕ β [ x β ( x, y, z, u, v ) , y β ( x, y, z, u, v ) , z β ( x, y, z, u, v )] , (35)with ∆ x,y,z = ∂ x + ∂ y + ∂ z and the effective potential v eα = v α ( x ) + l x α ( l x α + 1) x + l y α ( l y α + 1) y + l z α ( l z α + 1) z , (36)and h (4) αβ some integral kernels which can be found in [30]. For the case of 4identical bosons in the S-wave approximation former equations take the simpleform [29]( q + ∆ x,y,z − v ) ϕ ( x, y, z ) = v ( x ) (cid:20)Z − du xyx ′ y ′ ϕ ( x ′ , y ′ , z )+ 12 X i =1 , Z Z − dudv xyzx ′ y ′′ i z ′′ i ϕ i ( x ′ , y ′′ i , z ′′ i ) , ( q + ∆ x,y,z − v ) ϕ ( x, y, z ) = v ( x ) (cid:20) ϕ ( y, x, z ) + Z − dv xz ˆ y ˆ z ϕ ( y, ˆ y , ˆ z ) (cid:21) , where x ′ ( x, y ; u ) = 14 x + 34 y − √ xyu,y ′ ( x, y ; u ) = 34 x + 14 y + 12 √ xyu,y ′′ ( x, y, z ; u, v ) = 19 y ′ ( x, y ; u ) + 89 z + 49 √ y ′ ( x, y ; u ) zv,z ′′ ( x, y, z ; u, v ) = 89 y ′ ( x, y ; u ) + 19 z − √ y ′ ( x, y ; u ) zv,y ′′ ( x, y, z ; u, v ) = 13 y ′ ( x, y ; u ) + 23 z − √ y ′ ( x, y ; u ) zv,z ′′ ( x, y, z ; u, v ) = 23 y ′ ( x, y ; u ) + 13 z + 23 √ y ′ ( x, y ; u ) zv, ˆ y ( x, z ; v ) = 13 x + 23 z − √ xzv, ˆ z ( x, z ; v ) = 23 x + 13 z + 23 √ xzv. During the last twenty years these equations have been used to solve verydiverse bound state and scattering problems related with nuclear or atomicphysics, including the break-up reactions [31,32,33,34]. The interested readercan find in these references a detailed description of the proper techniques toimplement the boundary conditions and the associated numerical tools. As in the previous cases, the solution of the corresponding Schr¨odinger equa-tions starts with the Faddeev-like components associated to each interactingpair Ψ = X i 31 532 4 1 2 K H 53 4 1 2 5 1 2 3 45 T 45 3 S F z xyz w xyz w x xy zw yz w x y w Fig. 3 The same as Fig 2 for the N=5 case. Upper part: Tree diagram representing thefive possible topological ways for breaking the fully interacting 5-body cluster (12345) intosubstructures. Lower part: Jacobi coordinates corresponding to each cluster partition. Each of these ten Faddeev components Ψ ij is decomposed into six 4-bodylike Yakubovsky components Φ ij = Φ ijkij + Φ ijlij + Φ ijmij + Φ ij,klij + Φ ij,kmij + Φ ij,lmij , (38)defined, as in the N=4 case (31), by Φ ijkij = G ij V ij ( Φ ik + Φ jk ) ,Φ ij,klij = G ij V ik Φ kl . he Faddeev-Yakubovsky symphony 13 In its turn, the 4-body components are further decomposed in terms of fiveindependent 5-body FY components – denoted following [6,7] by K,H,T,S,F –in the following way: Φ ij = ψ ijkij + ψ ijlij + ψ ijmij + ψ ij,klij + ψ ij,kmij + ψ ij,lmij , (39)with ψ ijkij = K lij,k + K mij,k + T ij,k ,ψ ij,klij = H ij,kl + S ij,kl + F ij,kl . (40)Each of these 5-body FY components corresponds to a topologically indepen-dent way of breaking a 5-body cluster. At each step one of the ”interacting”clusters is decomposed into two pieces, giving rise to a corresponding FY equa-tion. The procedure is repeated until there remains only a single ”interacting”pair. The possible partition chains for a 5-body system are represented inthe upper part of Fig. 3. On the left side one may identify K-like and H-like structures, which are associated with 4+1 particle structures. The threecomponents on the right hand side (T-like, S-like and F-like) represent 3+2particle structures.The lower part of Fig. 3 represents the Jacobi coordinates associated toeach FY component. They have the general form (34) and are denoted by q = { x , y , z , w } . By permuting the particle index one can see that there are 60 differentK-type amplitudes, and 30 for each of the other types: H,T,S,F. Finally, thisresult into the set of 180 FY equations, first derived in [25],( E − H − V ij ) K lij,k = V ij h K lik,j + K ljk,i + ψ iklik + ψ jkljk + ψ ik,jlik + ψ jk,jljk i , ( E − H − V ij ) H ij,kl = V ij h H kl,ij + ψ klikl + ψ kljkl i , ( E − H − V ij ) T ij,k = V ij h T ik,j + T jk,i + ψ ik,lmik + ψ jk,lmjk i , ( E − H − V ij ) S ij,lm = V ij h F lm,ij + ψ lm,jklm + ψ lm,iklm i , ( E − H − V ij ) F ij,lm = V ij (cid:2) S lm,ij + ψ lmklm (cid:3) , (41)with ψ ’s given by (40).Each FY component F a =(K,H,T,S,F) is expressed in its own Jacobi setand take now the values on R and expanded in spherical harmonics for eachangular variable: F a ( x , y , z , w ) = X α f a,α ( x, y, z, w ) xyzw Y α (ˆ x, ˆ y, ˆ z, ˆ w ) , (42)where Y α is a generalized ”quadripolar harmonic” accounting for the angularmomentum, and eventually spin and isospin, couplings Y α = (cid:2) [ l x , l y ] l xy [ l z , l w ] l zw (cid:3) L , (43) and α labels the set of quantum numbers involved in the intermediate cou-plings.After projecting the angular part, one is left with a set of coupled four-dimensional integro-differential equations for the reduced radial amplitudes f a,α of the form (cid:2) q + ∆ xyzw − v eα ( x ) (cid:3) f a,α ( x, y, z, w ) = v α ( x ) X bβ Z Z Z dθdξdζ h (5) aα,bβ ( x, y, z, w, θ, ξ, ζ ) f b,β ( x b , y b , z b , w b ) , (44)with ∆ xyzw = ∂ x + ∂ y + ∂ z + ∂ w and the effective potential v eα ( x ) = v α ( x ) − l x ( l x + 1) x − l y ( l y + 1) y − l z ( l z + 1) z − l w ( l w + 1) w . These equations have been solved for the first time in a recent work devotedto study the n- He scattering [7], more recently they were applied to computethe complex energies of the H resonant states [8] In the last years we have developed a numerical protocol to solve the FYequations in configuration space for N=3,4 and 5 particles. For the sake ofsimplicity we will illustrate it in the case N=3; the close analogy we kept inthe final equations (22), (35) and (44), make the generalization rather straight-forward.In N=3, we wish to determine on a two dimensional domain D = [0 , x n x ] × [0 , y n y ] the partial wave amplitudes of the FY components which are solutionof eq. (22). Our unique approximation is the assumption that the solution weare looking for can be locally expanded in terms of some polynomial basis: ϕ α ( x, y ) = N x X i =0 N y X j =0 c α,ij f i ( x ) f j ( y ) . (45)We used two kind of such bases: the so called splines f i ( x ) ≡ S i ( x ) (cubicor quintic) and the Lagrange interpolating functions f i ( x ) ≡ L i ( x ). In whatfollows we will particularise the case of cubic splines. The interested readercan find a detailed explanation of both choices in Sect. 2.9 of Ref. [26].The cubic spline functions S i ( q ) are associated to each variable q = x, y ,they are constructed upon a ( n q +1)-point grid G q = { q , q , . . . , . . . q n } definedon each of the intervals [ q = 0 , q n ] of the resolution domain D .Two splines are associated to each grid point q i : S i and S i +1 . Both have afinite support in the two consecutive intervals [ q i − , q i ] ∪ [ q i , q i +1 ], are piecewise he Faddeev-Yakubovsky symphony 15 cubic polynomials on each of them and have C matching between them. Theyhave the useful properties: S i ( q j ) = δ i, j ; S ′ i ( q j ) = δ i, j +1 ; ∀ j = 0 , n q . By inserting (45) into eq. (22) and validating on an ensemble of N x × N y ≡ (2 n x + 2) × (2 n y + 2) well chosen points { ¯ x i ¯ y j } for each FC one obtains a linearsystem allowing to determine the unknown coefficients of the expansion (45): Lc = Rc ⇐⇒ X α ′ i ′ j ′ L αij,α ′ i ′ j ′ c α ′ i ′ j ′ = X α ′ i ′ j ′ R αij,α ′ i ′ j ′ c α ′ i ′ j ′ . (46)with i, i ′ = 0 , . . . n x + 1 and j ′ = 0 , . . . n y + 1 and α, α ′ = 1 , . . . n a where n a is the number of partial wave amplitudes of the Faddeev component φ a .Usually, by considering the fact that Faddeev components are regular at theorigin, the i = 0 and j = 0 splines might be neglected as they will requirecoefficients with null values. Then the dimension of the linear system for N=3is d = 3 × n a × (2 n x + 1)(2 n y + 1).The matrix elements of the left hand side are generalized by L αij,α ′ i ′ j ′ = δ αα ′ (cid:2) q S i ′ (¯ x i ) S j ′ (¯ y j ) + S ” i ′ (¯ x i ) S j ′ (¯ y j ) + S i ′ (¯ x i ) S ” j ′ (¯ y j ) − v eα (¯ x i ) S i ′ (¯ x i ) S j ′ (¯ y i ) (cid:3) , and those of the right hand side by R αij,α ′ i ′ j ′ = v α (¯ x i ) Z +1 − du h αα ′ (¯ x i , ¯ y j , u ) S i ′ [ x ′ α ′ (¯ x i , ¯ y j , u )] S j ′ [ y ′ α ′ (¯ x i , ¯ y j , u )] . In the scattering problems, the boundary conditions introduce an inhomo-geneous term in the right hand side of (46) and one is left with solving a linearsystem, generalized as Ax = b ; A = L − R, (47)where x holds for the unknown coefficients c αij . However the bound stateproblem is an homogeneous one Ax = λx, (48)where both λ and x are unknowns. Both problems can be unified by using theinverse iteration method. It consist in solving iteratively the inhomogeneoussystem ( A − λ ) x ( k +1) = x ( k ) ; k = 0 , , ....nite. starting with a trial value λ and an initial guess x . One can show that undersome conditions the series x (0) , x (1) , x (2) , . . . converges towards the eigenvectorof A whose eigenvalue is closest to the trial value λ . By doing so we are alwaysreduced to solving an inhomogeneous linear system.In view of solving very large linear systems Ax = b. (49) the direct methods (Gauss elimination, LU decomposition,...) are not appli-cable for they require storage capabilities beyond the present technology andturns to be very slow. We use alternative methods, based also on iterative pro-cedures which, starting from an ”educated guess” x (0) , generate a series x ( i ) which minimize the residual r ( i ) = | Ax ( i ) − b | until we consider it to be ”smallenough”. They are based on the matrix-vector multiplication operations whichrequire only to store a small amount of data, which allows reconstruction ofthe matrix A elements on the fly.There are several families of iterative algorithms [35]. The so called Bi-Conjugate Gradient Stabilized (BICGSTAB) is a very robust one that wehave extensively used.One of the limitation of the iterative methods is however the number ofiteration required until convergence. Even in the 3-body problem with d ∼ this number is prohibitive. To overcome this difficulty one uses the precondi-tioning technique, which consist in finding an approximation of the inversematrix A − , say ˆ A − , and solve instead of (49) the equivalent systemˆ A − Ax = ˆ A − b. (50)We have systematically used this technique taking as a preconditioning matrixthe one appearing in the left-hand-side of equations, the matrix ˆ A = L . Itsinversion can be performed exactly by means of the so called ”tensor trick”,introduced by [36]. It is based on the fact that the matrix L is a sum of 4 termshaving a tensorial structure L i = P i ⊗ Q i . We will illustrate this procedure inthe case of one single amplitude L ij, ′ i ′ j ′ = q S i ′ (¯ x i ) S j ′ (¯ y j ) − v (¯ x i ) S i ′ (¯ x i ) S j ′ (¯ y j )+ S ” i ′ (¯ x i ) S j ′ (¯ y j ) − l x ( l x + 1)¯ x i S i ′ (¯ x i ) S j ′ (¯ y j )+ S i ′ (¯ x i ) S ” j ′ (¯ y j ) − l y ( l y + 1)¯ y j S i ′ (¯ x i ) S j ′ (¯ y j ) , One easily identifies the following tensorial structure L = L x ⊗ N y + N x ⊗ L y , where the factors N xii ′ = S i ′ (¯ x i ) ,N yii ′ = S j ′ (¯ y j ) ,L xii ′ = q S i ′ (¯ x i ) + S ” i ′ (¯ x i ) − l x ( l x + 1)¯ x i S i ′ (¯ x i ) − v (¯ x i ) S i ′ (¯ x i ) ,L yjj ′ = S ” j ′ (¯ y j ) − l y ( l y + 1)¯ y j S j ′ (¯ y j ) , have dimensions usually equivalent to the square root of the original matrix L . Let us rewrite L it in the form L = N x ⊗ N y · ( N − x · L x ⊗ y + x ⊗ N − y · L y ) , he Faddeev-Yakubovsky symphony 17 and diagonalize in C N − x · L x = U x · D x · U − x ,N − y · L y = U y · D y · U − y , with U unitary and D’s diagonal matrices. We obtain L = N x ⊗ N y · ( U x · D x · U − x ⊗ y + x ⊗ U y · D y · U − y ) , which can be written in the form L = N x ⊗ N y · U x ⊗ U y · ( D x ⊗ y + x ⊗ D y ) · U − x ⊗ U − y , (51)The inverse matrix is then easily obtained L − = U x ⊗ U y · ( D x ⊗ y + x ⊗ D y ) − · U − x ⊗ U − y · N − x ⊗ N − y , (52)with ( D x ⊗ y + x ⊗ D y ) − ij,i ′ j ′ = ( D xi + D yj ) − δ ii ′ δ jj ′ . This procedure, that can be generalized to N=4 [37] and N=5 case, allowsus to obtain a satisfactory preconditioning of the linear system (49) and a finalsolution with a reasonably small number of iterations ( ∼ n x , n y , . . . )for each coordinate q = x, y, . . . as well as in the number of partial waveamplitudes ( n a ) on which the Faddeev-Yakubovsky components are expanded.The convergence of the results is studied by systematically increasing thesenumbers. We have attempted to describe, in a unified scheme of increasing complexity,the Faddeev-Yakubovsky equations for the N=3, 4 and 5-body problems inconfiguration space, as well as the numerical methods allowing its solution.They are based on a recursive splitting of the intrinsic N-body wave func-tion, solution of the Schr¨odinger equation, in to as many components as thereexist independent ways (or ”complete partition chains”) to break fully con-nected N-particle system into disconnected ones with a single interacting par-ticle pair remaining. They are represented by the ”tree” diagrams of Figs. 1, 2and 3. To each diagram is associated a function, named Faddeev-Yakubovskycomponent, which takes the values on R N − . The pioneering work of L.D.Faddeev for N=3 [1] and its extension to an arbitrary N achieved by O.A.Yakubovsky [5], both from the Leningrad/Saint Petersburg University, formu-lated the equations fulfilled by the ensemble of these components.The original works of L.D. Faddeev and O.A. Yakubovsky were formulatedin momentum space, in terms of T-matrix partitions, and solved under this form by Gl¨ockle and collaborators for N=3 and N=4 [40] The formulationin configuration space was due to S.P. Merkuriev and S. Yakovlev in a fruit-ful collaboration with the Grenoble theory group of the former Institut desSciences Nucl´eaires [9,10,12,13,27,28,29].In configuration space, the Faddeev-Yakubovsky equations result into asystem of partial derivative equations coupling the ensemble of the Faddeev-Yakubovsky components. After the partial wave expansion is performed, theyturn into a coupled system of integro-differential equations on R N − with somesmooth (N-2)-dimensional integral kernels which can be solved by standardlinear algebra methods. Their generalization is natural, as it can be seen from(22), (35) and (44)The Faddeev-Yakubovsky equations provide a mathematically rigorous ap-proach for the full solution of the N-body problem. However its scalability withthe number of interacting particles is a serious drawback that dramaticallylimits its range of applicability. The fact that the formal object to study isa function with arguments on R N − is a first sign of it, but not the onlyone. The situation, discussed in some detail in [6], is well illustrated in Table1 where the dimension of the N-body solution is detailed in terms of numberof equations, partial wave amplitudes, and the linear algebra problem.We would like to notice that other rigorous schemes have been proposedfor the solution of the N-body problem. One of them is the AGS equations [38]which have provided accurate results for the 3- and 4-nucleon problem [39].It is worth emphasizing also that such a rigorous mathematical schemesare not necessary when dealing with bound states or simple 1 + ( N − 1) elas-tic scattering processes. The Schr¨odinger equation can then be directly solvedby several methods, like GFMC [41], with Hyperspherical Harmonics [42,43]or Gaussian basis [44], NCSM [45,46], Lorentz Integral Transform [47] whichproduces also very accurate results, in some cases well beyond the technicalcapabilities of the Faddev-Yakubovsky approach (see [48] for a more detailedreview). However, the Faddeev-Yakubovsky partition of the wave function isinteresting to increases the numerical convergence of the results or even un-avoidable for an appropriate implementation of the boundary conditions [44,49]. References 1. L. D. Faddeev, JETP 39, 1459 (1960).2. L.D. Faddeev, Sov. Phys. JETP 12, 1014 (1961).3. L.D. Faddeev, Mathematical aspects of the three-body problem in the quantum scatteringtheoryIsrael Program for Scientic Translations, Jerusalem 1965.4. L.D. Faddeev and S.P. Merkuriev, Quantum Scattering Theory for several particle sys-tems, Kluwer Academic Publishers (1993).5. O.A. Yakubovsky, Sov. J. Nucl. Phys. , 937 (1967).6. R. Lazauskas, Few-Body Syst 59, 13 (2018).7. R. Lazauskas, Phys. Rev. C 97, 044002 (2018).8. R. Lazauskas, E. Hiyama, J. Carbonell, Phys. Lett B791, 335 (2019).9. S.P. Merkuriev, Theoretical and Mathematical Physics 8, 798 (1971).he Faddeev-Yakubovsky symphony 19 Table 1 Scalability of the Faddeev-Yakubovsky scheme as a function of the number ofinteracting particles N.: ne is the number of equations,(equal to the number of FY compo-nents), neid the same number in case of identical particles, n a the number of amplitudes inthe partial wave expansion.N Ne Neid n a ∼ ∼ ∼ N !( N − N − Int (cid:16) N − π/ N (cid:17)(cid:17)