Asymptotic equations for two-body correlations

Asyptotic equations for two-body correlations M. Fabre de la Ripelle Abstract. An asyptotic equation for two-body correlations is proposed for a large nubers of particles in the frae work of the Integro-Differential Equation Approach. The quality of this equation is discussed with exaples. Adiabatic and asyptotic properties of the two-body correlations are investigated. Keywords : Nuclear Structure, Hyperspherical Haronics, Integro-differential Equation. e-ail address : ineko13@hotail.co Adress for correspondence : M. Fabre de la Ripelle, 13 rue de l Yvette, 7516 Paris, France Introduction The integro-differential equation approach (IDEA) was proposed in 1984 [1] to avoid solving the large nuber of coupled differential equation of the Hyperspherical Haronic Expansion Method (HHEM) needed to obtain an accurate converged solution for nuclei beyond systes of few bodies []. It already included the asyptotic expression of the projection function, which is the ain ingredient in the equation for a large nuber of particles [1], and analytical solutions for bosons in an S-state were also derived [3]. In a recent paper [4] another derivation of the asyptotic equation has been published for bosons in the ground S-state. In order to obtain this asyptotic equation, the autor correctly deduced an equation (3) extracted fro ref. [5]. But contrary to their allegation this equation cannot lead to their asyptotic equation (5). The asyptotic equation can be obtained directly fro the asyptotic projection function already published in ref. [1], a reference quoted in [6].The ai of the present paper is to establish the correct equation and to discuss the conditions for the validity of its utilization, as a substitute for the exact equation, for a large syste of particles. The ethod used in this paper to solve the Schrödinger equation V( x) E ( x ) where A 1 i is the Laplace operator in the whole D 3 A diensional space for A particles, V( x ) a function of all coordinates x( x1,..., x A) and E the eigenvalue related to the wave function ( x ), proceeds step by step. First by assuing V = E =, the solutions are ( x ) = (Haronic polynoial in the x coordinates Y ( ) [ ] x ). Since Haronic polynoials are hoogeneous polynoials the radial coordinate r with r² x ² can be factorized and ( x) Y ( ) r L written in polar coordinates (, r ) where Y ( ) [ ] is a Hyperspherical [ L] Haronic defined by the D-1 quantu nubers [L] including L the degree of the polynoial. L L i 1

Introducing a Hypercentral Potential Vr, () i. e. depending only on the radial coordinate r, leads to a solution Schrödinger radial equation. ( ) ( ) ( ) / ( D 1)/ x Y[ L] u r r where () ur is an eigenstate of a standard In nuclear physics, ore than 3/4 of the potential energy originates already fro the Hypercentral part (invariant under rotation in the whole space of coordinates) of the nuclear interaction, and balances the contribution of the kinetic energy to the binding energy. To go further the structure of V( x ) ust be known. For a su of two-body potentials V ( x) V ( r ), r x i x j the wave function i, j i ( x) Y ( ) P( r, r ) is a su over all pairs of two-body aplitudes P( r, r ) solutions of [ L] i, j i an Integro-Differential Equation (IDE) [1]. Traditionally the any-body bound states are defined by assuing that a state can be written as a properly syetrised product of individual particle eigenfunctions of a one-body potential well, and a Jastrov function. This Jastrov function is the product over all pairs of variational functions f ( r ), r xi x j, where the pair variational function is selected to gives the strongest binding energy. This procedure suffers of several defects : 1/ Absence of an independent center of ass for a pair wise interaction. / Inability to define the wave function out of the range of the potential, a defect of all variational ethods. 3/ Inaccuracy in the coputation of the wave function where it is sall, since only the square of the wave function is significant in the variational procedure. 4/ Absence of an equation to define the correlations. In contrasts another ethod has been proposed where, assuing the analyticity of the wave function, the state is defined by the polynoial of the lowest degree L occurring in an haronic polynoial expansion of the wave function [1]. For bosons where all particles can be in 1s state this degree is L. For identical Ferions it is given by the lowest degree antisyetric polynoial hoogeneous in the individual coordinates x ( x1,..., x A) of the A-particle syste, where the antisyetry is provided by a Slater deterinant. This polynoial is translationally invariant since any scalar syetrical operator seeking to decrease the degree of a hoogeneous polynoial should give zero when applied to a polynoial of inial degree.

Since the state is defined by a translationaly invariant haronic polynoial [9], therefore it can be expressed in ters of Jacobi coordinates. The radial equation Let D ( ) [ ] x be the antisyetric haronic polynoial written as a Slater deterinant. L In a polar coordinates syste x ( r, ) where A r² ( x X )² and X is the center of ass, is a set of D 1 angular coordinates in the D 3( A 1) diensional space of the Jacobi coordinates. We can then write L D ( ) D ( x) / r is a Hyperspherical Haronic (HH). (1) L L ( D 1)/ L (, r) D ( ) ( ) / L u L r r () The state is defined by the product of a HH and a (Hyper) radial wave solution of the radial equation ² d² L ( L +1) V[ L ]( r) U ( r) E u ( ) L r, L L ( D 3) / (3) dr² r² where V () [ ] r is a Hypercentral potential. L When the interaction is a su over all pairs of a two-body potential V( r ), the Hypercentral potential V () [ ] r is a su over all pairs of the rotationally invariant part of the pair-wise L potential while U () r is the contribution of the pairs correlation. 1 i V ( r) A( A 1) / D ( ) V( r ) D ( ) d (4) [ L ] [ L ] [ L ] Contribution of the two-body correlations This contribution is the eigenpotential in a two-variable Integro-Differential Equation obtained by writing that, for a su of pairwise potentials, the wave function is the product ( r, ) D ( ) P( r, z ), (5) [ L ] k A k where z r r for r / r cos, k cos k k / ² 1 kl kl of the state and a su over all pairs of two body aplitudes [1]. Without correlations U and the solution is given by () and (3). 3

As A increases an iportant decoupling occurs between the variables z and r, and a good approxiation is obtained by taking for the solution the product (for P( r, z) P( z, r) u( r ), (6) z r r ) for a reference pair (i, j), where P (, ) r z is an eigenfunction for each r (r / ² 1 is a paraeter) of the Integro-Differential Equation (IDE). 4 ² 1 d d 1 z (1 z²) W[ L] P ( z, r) ( V ( r V ( r))( P ( z, r) r² W ( z) dz dz 1 1 [ L] f ( z, z ') P ( z ', r) dz ') U ( r) P ( z, r) (7) with [ L] [ L ] where V ( ) r is the integral in (4), taken over all Hyperangular coordinates while u ( ) r becoes a solution of (3). The projection function reference pair ( i, j ). It is given by [1]. f ( z, z ') projects the aplitudes P (, ) r z in (5) on the space of the k f z z W z f P z P z (8) [ L] [ L] (, ') [ L] ( ') ( K 1) K ( ) K ( ') K where the [ L P ] ( z ) are the noralized polynoials associated with the weight function K W ( ) [ L] z while the coefficient are [1] [ L] ( A 3) [ L] [ L] fk 1 ( A )( PK ( 1/ ) PK ( 1)) / P K (1) (9) 4 The weight function is [7, 8] W z z z Q z with L ( D 5) /. Q( z) D[ L] ( ) ² d 1 1/ [ L] ( ) (1 ) (1 ) ( ) where the integral is taken over all Hyperangular coordinates except z is a polynoial of degree where refers to the last filled shell in nuclei with,1,,... for the s, p, s-d shells. The asyptotic equation For bosons in the ground state L, Qz ( ) 1 and the associated polynoials are the Jacobi polynoials P,1/ ( z ). The weight function is K 4

W ( x) (1 x² / r²) x ² (1) in ters of x r with dz ( x ) function as. 4x dx. where for a noralized W( x ) the first ter evolves into a r² Analytical expressions for the su in (8) have been already found : first in the case of threebodies where it leads to a Faddeev equation [9] and then for a large nuber of bosons when [1]. We now look back to details that were involved of the last calculation, to analyze the quality of the approxiations. The following are the ain points : 1 / One uses the asyptotic relation li 1 ² / Z ² Z e (11) this approxiation is valid for Z ² / The polynoials associated with the noralized weight function W Z dz e Z dz (1) ( ) 4 / Z ² ² [] are the Laguerre polynoials 3 / L 1/ K ( Z ²) It can be proved easily fro the asyptotic properties of the Jacobi polynoials used in ref [3] for z, n and fixed, 1 z 1 n n! n,, li Pn ( z), Pn (1) and z 1 in Eq. (9) that asyptotically the coefficient n n is a binoial coefficient, for z 1/ f 1 becoe [1, Eq. (9.6)]. K li 1 ( ) / 4 K K f A (13) 4 / If all the asyptotic relations are valid the forula [11] 5

n! S Ln( x) Ln( y) z ( n 1) n 1 x y 1/ ( xyz) (1 z) exp z ( xyz) I 1 z 1 z n 1/ (14) can be used for z 1/ 4, 1/ associated with the weight function x xe. Here one sets Z² x² r² with 3 A / A in (1) (15). The radial function ur () is concentrated around the iniu at potential ² L ( L 1) Veff ( r) V[] ( r) U ( r) r² r r of the effective (16) d in Eq. (3) where V eff ( r ). dr The r..s. radius a is related to r by r² r² Aa ² (17) and the validity of Eq. (11) is fulfilled for x r Aa. In the ground state, all bosons in ² ² the 1s state have the tendency to stay inside the range of the potential, in such a way that the r..s. radius should be either stationary or vary very slowly with the nuber of particles. Eq. (7) when solved exactly showed a stability of the r..s. radius a for growing A. For instance with the Afnan-Tang S3 potential [1], the r..s. radius a 1.34.1f between A 6 and A 16 and a 1.4 f for A 4 [7]. The behavior of the coefficients (9) with respect to K ust now be investigated. In table 1 the product K. f A is exhibited for A 16,4,1,5,1 for K 4 ( K 1) / ( ) 6

Table 1 K A 16 4 1 5 1.788.9149.9663.9933.9966 3 -.789.4771.791.958.979 4.87.34.6711.99.9643 5 -.146.195.534.8934.9458 For K we have to deal with the weight function approxiation where the projection function is the weight function itself, with f A( A 1) /. The K 1 ter generates a spurious coponent which ust be eliinated [13]. It is clear fro the table 1 that 4 K over estiates the contribution of the ters of degree K in the projection functions. But we have shown previously [6] that nearly all the contributions to the binding energy originate fro the few first ters in the expansion (8). Since the weight function (1) with (15) is defined in ters of x r it is appropriate to use the Integro-Differentiel Equation in the relative coordinates version [3] i.e. with the sae variable x. The su (14) with 1/, z 1/ 4, x Z², y Z '² is S ( Z² Z'²)/3 4 e 4 sinh( ZZ ') 3 ZZ ' 3 (18) (note that in [4] the forula (15) of ref. [5] which gives 3/4 instead of 4/3 is flawed, indeed a ter t 1/ is issing in the last parenthesis). It leads to the projection function [1] 3 3 f ( z, z ') dz ' ( A ) / A 3 Z Z ' e 3 4 1 3 ZZ ' ( Z ' Z ) /3 ( Z ' Z ) /3 e e Z ' dz Z ' ' (19) which can be inserted as it is into (7) with Z² (1 z ) / (note dz ' 4 Z ' dz '/ ). 7

It can be easily transfored as well into an equation in the variable Z by using Eq. (38) of ref [3] which eliinates the first derivative in the variable x r in the kinetic energy operator by writing P( z, r) g / g with x² x² g x 1 (1 ) Ze r r² r² / Z ²/ () and thus P z r e Z G Z r. By introducing this transforation together with the (, ) Z ²/ / (, ) asyptotic weight function (1) in Eq. (7) one generates the asyptotic equation in the variable Z. The kinetic energy operator becoes Z ²/ 4 ² 1 d d ² e d² r W z dz dz r Z dz (1 z²) W P ( z, r) Z² 3 G ( Z, r) ² ( ) ² ² (1) And the wave equation (7) ultiplied by r Ze Z ²/ is transfored (with (19)) into ² ² d² r² Z ² 3 G ( Z, r) V ( Z, r) U ( r) G ( Z, r) dz ² ² 1 Z '²/ Z ²/ e r² VZe f ( z, z ') G ( Z ', r) dz ' ² Z ' 1 () One introduces Z ²/ e inside the integral to produce an equation where the r. h. s. becoes r ² V F ( Z, Z ') G ( Z ', r ) dz ' (3) ² The projection function 3 3 F( Z, Z ') ( A ) / A 3 ( Z² )( Z '² ) ZZ ' e 3 4 e 3 (5( Z Z ')² ZZ ')/6 (5( Z Z ')² ZZ ')/6 e ( Z² Z'²)/ is syetrical in Z and Z, V( Z, r) V( r) V ( r ) for r r / Z, V ( ) r is the hypercentral part of V( r ) and the boundary conditions G( o, r) G(, r ) deterine U () r. Applications with coents In the Hyperspherical Approach we obtain the wave function and binding energy fro two sei independent equations : first fro an equation where the contribution of the two-body 8

correlations to the binding energy appears as an eigen-potential U () r of the radial coordinate. This potential is then introduced in a radial equation generating a radial wave function and a total binding energy. The Hypercentral potential V () [ ] r occurring in the radial equation can be easily coputed fro the original weight-function. L But in the other equation () only the residual potential V( r, r ) where the Hypercentral part of V( r ) has been subtracted ust occur. This Hypercentral part is calculated fro the asyptotic weight function which differs fro the exact one. Therefore an inaccuracy, in the calculation of the correlations, originates first fro the asyptotic weight function and then fro the asyptotic projection function itself. In the projection function, when the nuber of ters needed to reach a requested accuracy for U () r is sall, only the defect of the asyptotic weight function with respect to the original one should have an effect. Here one presents a few coputations for A 16 and 4 with typical nuclear potentials. The IDE can be easily and accurately solved nuerically with a progra [8] that uses the traditional algorith to integrate for the Independent the Particle Model the radial Schrödinger equation (see ref. [14]). Four potentials are chosen with an increasing strength of the repulsive core: the Gogny, Pire, de Toureil (GPDT) [15], Brink-Boeker B1 [16], Afnan-Tang S3 [1] and Malfliet-Tjon MTV [17] potentials. First one recall a few results obtained with the Volkov [18], Afnan-Tang S3 [1] and Malfleit- Tjon MTV [17] potential for A=16 bosons, for which the binding energy were calculated with the Hypercentral Approxiation (HCA), the Weight Function Approxiation (WFA) and the IntegroDifferencial Equation Approach (IDEA). Table Pot. S3 Volkov MTV Ref. [19 ] [] [3] [7] [19 ] [3] [19 ] [] [3] HCA 799 797.67 797.5 798.56 156 1559 74 737.77 737.5 WFA 131.4 131.8 166 165.8 136 1363.9 IDEA 135 135.13 135 135.45 163 163 1363 136.7 1363 It is clear that the contribution to the binding energy, of the polynoials for K in the projection function, does not exceed a few Mev. Thus only the WFA, where the ter K in the projection function is the weight-function itself, is actually significant. 9

Four different algoriths have been chosen to solve nuerically the any body bound state proble. In ref. [19, ] the eigenfunction P( z, r ) is expanded in ters of soe interpolating polynoials, thereby reducing the equation to a standard eigenvalueproble. The B splines of order N have been chosen for this purpose. In ref. [] the Hyperspherical Haronic Expansion Method (HHEM) was applied using up to 4 coupled equations. In ref. [3] the solution of the I. D. E. is discretized by using a finite difference schee and an appropriate inverse power ethod provides the requested eigenvalue by the diagonalisation of a large atrix. In ref. [7] and in the present paper the I. D. E. in the variable x r of ref. [3] is solved nuerically by generating the aplitude with a standard shooting procedure [14] : the solution of g "( x) ( W ( x) U ) g( x ) is generated by g( x h) g( x) g( x h) ( W ( x) U ) h ², for an interval h between two successive esh points with the boundary conditions g() g( x r ) reached for the eigenvalue U () r. The slope of the shoot g( h) Ph is chosen in order to noralize P( x, r ) to unity i. e. P( x, r) 1 h( x, r ), details are available in ref. [8]. All ethods agree surprisingly well. The variational ethods of ref. [3] provide for the S3 Potential a binding energy of 113.94 Mev fro the Translationally Invariant Configuration Interaction (TICI) ethod and 134.86 Mev for the Translationally Invariant Coupled-Cluster (TICC) ethod. In the last ethod the algebraic expression to be solved nuerically covers ore than one full page in ref. [3] while it takes one line in this paper! The converged value for the binding energy is already reached with only four polynoials in the projection function, and the weight function approxiation (WFA) for K gives an estiate of the binding energy with an accuracy better than half percent at the level of twobody correlations. The WFA with K gives the largest increase of binding energy with respect to the Hypercentral Approxiation, without correlations, where U ( r ). On the other hand the WFA is independent of the polynoial occurring in the projection function since the first polynoial for K is P 1. Whatever the weight function used and when the aplitude P( z, r ) is noralized to one the integral in (7) becoes A(A-1)/-1 obviously independent of r. Solving then the WFA, one test first the effect of the asyptotic weight function (alone) on the solution of the IDEA. 1

Near the iniu, at r of V () r where dv ( r) / dr, the effective Hypercentral potential behaves like an Haronic oscillator and the. s. radius a r A r r r A. ² ² / ( ( )² ) / eff eff The r.. s. radius a is given table 3 together with a r / A. Table 3 A 16 4 Pot. GPDT B1 S3 MTV GPDT B1 S3 MTV a 1.4 1.7 1.36 1.3 1.4 1.3 1.45 1.41 a 1.1 1.4 1.3 1.8 1. 1.8 1.4 The very sall difference between the a and a supports the estiate in eq.(17) and the condition for the validity of (11), assuing that the strong concentration of the radial function near the iniu of the effective potential is fullfilled. The fully asyptotic binding energy obtained fro eqs (, 3) with 3 A / in the Asyptotic Kinetic energy and in the Asyptotic Weight Function is exhibited in table 4. In the first line the exact WFA is given, then the exact Hyper Central Approxiation HCA where U ( r ), and next the sae for the fully asyptotic energies where 3 A / in Eq. (). Finally in the last line one finds the cobination Test = exact HCA Asyptotic HCA + ASWFA, where ASWFA is the binding energy given by the Asyptotic Eq. (). Table 4. Contribution of the fully asyptotic approxiation to the binding energy of 16 and 4 bosons in the ground state, AS is for asyptotic. Table 4 A 16 4 Pot. S3 B1 GPDT MTV S3 B1 GPDT MTV WFA 13.1 168.8 187.8 1363.4 7835.6 1176 96 85 HCA 798.3 146.6 1149.6 738.6 6315.4 191 943 5837 ASWFA 15 1658 179 1345.6 777.5 1165.8 9517.4 843.8 ASHCA 78.7 1436 1141 76.3 618.5 1787.9 8957.7 5757.5 TEST 1.6 168.6 187.6 1357.9 784.4 1177.9 96.7 853 A coparison between the Test line and the exact WFA energy shows clearly that the asyptotic Eq. () provides a rather good estiate of the contribution of the correlations to 11

the binding energy but that in order to obtain a binding energy siilar to the one of the WFA, the HCA ust be calculated with V () [ ] r obtained fro the original weight function in the radial equation (3). L Nevertheless, the larger the difference between WFA and HCA, the larger is the difference between the test line and the exact WFA. As a conclusion, the asyptotic weight function can be used to calculate U () r, the contribution of the correlations to the binding energy, but in the radial equation the original weight function should be used to calculate V () [ L ] r. One ust stress that the calculation of U () r ust be done with the asyptotic projection function, by starting fro the residual potential V V( r ) V ( r ) where the Hypercentral potential V ( ) r is calculated with the asyptotic weight function. Nevertheless one cannot avoid keeping in ind that for systes, like bosons in the ground state, where all particles interact together within the range of the potential, the any body correlations, and at first the three and four body correlations, have to be taken into account [1]. For nuclear systes however, the increasing of iportance of the high-order correlations is expected to be uch less relevant, due to the role played by the Pauli principle [1]. Another ethod can be used to generate asyptotic equations. Our purpose is to substitute asyptotic expressions for the original equation, in the case of a large nuber of particles. For this task it is easier to use the IDEA equation in ter of x r developed in ref. [3] and reproduced in Appendix. In this equation the kinetic energy ter appears as g"/ g where x² g x(1 ) P ( x² / r²), / [ ] r² L 1, and P [ L ] is a polynoial occurring for Ferion systes while P [] 1 for bosons in their ground state and ( D 3) / L, with L for bosons. The ter g "/ g ( 1 ( ) / T) / T, (4) r² with T 1 X for X x² / r ² becoes a hard repulsive wall for x r, indeed with x r x. 1

( ) ( ) r² T ² 4( x)² x r (5) behaves as a strong repulsive kinetic centrifugal barrier for x r. x² One assues in the asyptotic approxiation that the significant values of occurring in r² the calculation of the wave function decreases for increasing nuber of particles which justifies the exponential approxiation X T e (6) When the first order expansion 1 T (1 X X²...) (7) is used and n X is neglected in (7) for n 1 the kinetic energy ter (4) becoes ( 5) X 3 O( X ²). It is an Haronic oscillator potential in x² / r ² when X ² can be r² neglected. In the next table various approxiations of the kinetic energy are tested for A 16 and A 4 bosons where H, AH, T 1 and T refer to Haronic oscillator (HO), Asyptotic H where 3 A /, T 1 for 1 T 1 X and T for 1 T 1 X X ² in Eq. (4) with X x² / r ² and exact without approxiation. The coputation is ade with a space between esh points of.1 f for x and.5 f for r. A S3 B1 GPDT MTV HO 17.5 1673.7 177. 1355.9 AHO 13.7 1677.6 18.8 1364.1 16 T1 13.5 1678.4 181.6 1364.1 T 131.7 1681.7 186.4 1363.1 Ex. 13.1 168.8 187.8 1363.4 HO 789.5 11718.8 9591. 853.8 AHO 783.8 117.6 9593.3 855.3 4 T1 7834.5 117.6 9595.3 855.7 T 7835.1 1175.9 9599.6 854.3 Ex. 7835.6 1176.1 96. 85.5 13

It is worthwhile to notice that the asyptotic AHO approxiation is very siilar to the T1 approxiation with a difference of around 1 Mev, or less, for A=16, and Mev for A=4. The good accuracy of the nuerical progra, each coputation being independent, enables one to check the regular iproveent between two successive approxiations. The next asyptotic approxiation to be analyzed concerns the weight function in Eq. (A) in Appendix with g T (1 X ), X x² / r ² for bosons. In the asyptotic approxiation X (1 X ) e f ( X ) (8) where f ( X ) (1 X ) e X and f( X ) 1 for an asyptotic exponential function. The polynoial expansion X 3 4 (1 ) (1 ² / / 3 / 8...) X e X X X (9) leads to an approxiate forula 3 X X X Z ²(1...) 3 8 (1 X) e (3) X x² / r ², Z² x² r² liited to the third degree ter. The ter of degree zero is for an asyptotic Gaussian weight function associated with Laguerre polynoials. In the table 5, the convergence to the exact value is exhibited in ters of N, the degree of the polynoial in X in Eq. (3). N= for a Gaussian asyptotic weight function. 14

Table 5 Potential S3 B1 A N WFA HCA WFA-HCA WFA HCA WFA-HCA 1151.1 73.16 4.85 161.71 137.31 31.4 1 11.5 79.45 431.5 167.31 1449.4 3.7 13.5 797.15 433.35 1681.1 1458.79.33 3 131.3 797.6 433.61 1681.81 1459.51.3 Ex 13.8 798.3 433.79 168.79 146.55.4 16 Pot GPDT MTV 1.3 171.81 15.51 18.94 679.7 63. 1 179.6 114.1 139.5 135.45 731.79 6. 186.54 1148.17 138.37 136.39 737.55 64.84 3 187.7 1148.7 138.3 1363.14 737.96 65.18 Ex 187.84 1149.63 138. 1363.4 738.56 64.84 Potential S3 MTV 7674.7 616.6 151.1 839. 576.3 685.9 1 788.3 637.7 15.6 851 583.5 69.5 4 7835.7 6314.9 15.6 853.5 5836.7 686.8 3 7834.9 6315.1 1519.7 853.7 5836.9 686.8 Ex 7835.6 6315.4 15. 85.5 5837.1 685.4 The difference between the approxiate binding energy for N 1 and the exact value is about 1 Mev i. e. less than one percent and decreasing for increasing A. The increase of binding energy WFA-HCA brought by the correlations is very siilar for each potential for N >, in such a way that a nearly exact binding energy can be obtained by adding this increase to the exact HCA. When N = the sae procedure isses the binding energy by about 1 Mev, and even ore for A = 16 and the MTV potential. The IDEA is a two variables integro differential equation with one length r and one angle defined by cos r / r, r xi x j related to the description of the two-body correlations of a bound syste of particles where pairs are in s-state. This equation can be reduced to two one variable equations by assuing that the aplitude P( z, r ) can be written as the product P( z, r) u( r) P ( z, r ) where P ( z, r ) is only 15

adiabatically r-dependent in such a way that the derivatives of P with respect to r can be neglected. This approxiation was first suggested for Atoic syste [4] and then extended to Nuclei [7, 8, 9]. An upper and a lower bound to the binding energy can be derived in this adiabatic approxiation [7, 8, 9]. The interval between upper and lower bound is less than about 1 Mev for 4 He, decreasing to.5 Mev for 16 bosons [3] and. Mev for A = 4, in such a way that by taking the iddle of the interval for binding energy reduces the inaccuracy by half. It vanishes as A. When Eq. (7) is ultiplied by r² the kinetic energy operator becoes independent of r and the variation with r of the kinetic energy should be adiabatic like P ( z, r ). Therefore one could expect that the eigenpotential r² U ( r ) ight also vary adiabatically. A potential where all the binding energy is generated by the correlations in nuclei was chosen for testing the behavior of the eigenpotential U in ters of r. The Argonne 14 potential [31] has a strong repulsive core which prevents any binding for both 16 O and 4 Ca at the level of the Hyper Central Approxiation without correlations (i. e. HCA > ). Since the equation (7) does not include the treatent of the tensor forces, the Afnan-Tang S1 potential [1] which is adjusted to give the experiental triplet even 3 S 1 Nucleon-Nucleon scattering phase-shifts is 3 substituted for the V tensor force of the original Arg 14 potential. For bosons in ground state the Afnan-Tang S3 potential [1], already used in nuerous test cases, is selected. For the sake of coparison, the table is divided in three coluns. In the first one U () r is ² exhibited followed by the ratio U ( r) / Kin( r ), where Kin( r) L ( L 1) / r² is the Kinetic Centrifugal barrier in the radial equation, and then by the effective potential V () r in the radial equation. eff 16

Pot S3 Bosons A = 16 A = 4 r (f) U ( Mev ) U / Kin Veff Mev U ( Mev ) U / Kin Veff 4 146 1.18 4758 6 748 1.41-84 8 437 1.46-181 3191 1.49 7183 1 88 1.5-14 36 1.68-516 1 5 1.54-75 178 1.8-7817 14 155 1.59-548 1341 1.9-6651 16 1 1.63-46 17. - 6651 18 99 1.67-36 87.6-556 Mev Pot. Arg. 14 Ferions 16 O 4 Ca r (f) U ( Mev ) U / Kin V ( Mev) r (f) U ( Mev ) U / Kin V ( Mev ) eff eff 8 54.71 198 774.54-17 1 38.664-1 636.538-8 1 5.636-144 4 533.537-338 14 149.68-13 6 453.536-346 16 114.69-17 8 39.534-33 18 91.634-85 3 338.533-34 74.64-67 3 96.531-74 6.647-53 34 6.59-45 4 53.654-43 36 33.58-17 6 45.66-34 38 8.56-191 8 39.669-8 4 187.54-169 The ratio U / Kin vary slowly, adiabatically, for growing r. The larger L, the better is the adiabatic approxiation. The ost significant ratio is in the vicinity of the iniu r of 17

V () r, in the doains of a few feris where the radial function ur () is not negligible, where eff it does not change by ore than a few percent or less. In this range the effective potential can be written ² Veff ( r) (1 U ( r) / Kin( r)) L ( L 1) / r² V ( r) where the ratio U / Kin is negative and nearly constant. An iportant difference appears between bosons and ferions systes. While the kinetic centrifugal barrier is repulsive for nuclei with the ratio U / Kin > -1 it becoes strongly attractive for bosons with a ratio < -1 generating a strong attraction in r and thus a collapse of the bosons which gather near the origin. The siilarity between the eigenpotential U and an attractive r potential has double effect. When a strong repulsive core is introduced in a potential to increase the size of a nucleus, it create a large aount of correlations which decrease the kinetic centrifugal barrier and then balance the effect of the core. For this reason the odified Arg. 14 Potential used in this paper which produces (without Coulob potential) a IDEA binding energy of 18.5 Mev and 336 Mev for 16 O and 4 Ca respectively [], in good agreeent with the experiental data, generates a r..s. radius of.5 f and.91 f respectively, saller than the experiental data. The adiabatic property of U / Kin suggests a change of variable where instead of x r in Eq. (A3) a new variable defined by x / r X / L with L L 1/ is chosen. The sall difference respect to L for growing A. L L ( L 1) 1/ 4 and 1 L becoe rapidly negligible with The derivative d² / dx² ( L / r)² d² / dx ² while the ter T in the kinetic energy Eq. (4) becoes 1 X ² / L ² independent of r. One renoralizes the residual potential V( r, r ) and U () r in Eq. (A3) according to V ( X, r) V( r, r) / Kin( r) L UL ( r) U ( r) / Kin( r) g / g ( X ² 3 L ) / r² (see Appendix) for bosons in ground state. One divides " Eq. (A3) by L ² / r² to obtain the asyptotic adiabatic equation. 18

d² dx ² ( X ² 3 L ) / L ² ( V ( X, r) U ( r)) G ( X, r) V ( X, r) G ( X, L ) I ( X, r) L L L L L Where r is a paraeter and G ( X, L ) is deduced fro Eq. (A) by substitution of X / L for x/ r with z x² / r² 1, L and a siilar transforation for I to obtain I L. The behavior in ters of r of the eigenpotential UL () r is adiabatic. Coents and conclusion The asyptotic equation (, 3) were obtained first by a transforation where the basic equation (7) is expressed in ters of the relative coordinate x r including an eliination of X the first derivative d / dx [3]. Using then the asyptotic expression (1 X) e, together with the relation (14), to su the series of Laguerre polynoials occurring in the projection function, Eq. (8) leads to an analytical expression (19) [1]. This expression can be used directly in Eq. (7) or transfored into Eq. () in the variable Z x / r, by a change of scale. With respect to the calculation published in ref. [4] : the forula chosen to su the series of Laguerre polynoials is taken fro ref. [5] and is used correctly. It leads to a coefficient 3/4 in Eq. (3) of ref. [4] instead of the 4/3 of ref. [11] and in our Eq. (18) and cannot lead to our asyptotic eqs (,3). The values quoted as exact, for the S3 potential and for 16 bosons in table of ref. [4], are not in agreeent with the five published values presented in table in this paper, which are in coon agreeent. We found, by application of the asyptotic equations, that for 16 and 4 bosons a rather fair binding energy can be obtained by adding the contribution U () r of the correlations, calculated fro the asyptotic equation, to the exact hypercentral potential V ( ) r, in the radial Eq. (3). It is, therefore, the asyptotic hypercentral potential which ust be introduced in the asyptotic equation to calculate U () r. We have shown that, thanks to the adiabatic approxiation, U () r behaves like a hypercentral kinetic energy centrifugal barrier with a ratio r attractive potential which reduces the strength of the U ( r) / / ( L( L 1)) / r which is -1 for ferions and -1 for bosons. It generates a reduction of the strength of the kinetic energy and of the r.. s. radius for bosons, producing a collapse, where all bosons gather inside the range of the potential. 19

This difference between bosons and ferions originates fro L in Eq. (3) which grows as 3 A / for bosons in the ground state ( L ) and as 4/3 1/ (3 / A ) for nuclei [8] which akes the ratio saller in absolute value for nuclei, than for bosons in the ground state. In the asyptotic radial equation the eigenpotential energy U () r behaves like an attractive kinetic energy ter, while the kinetic energy ter in the aplitude equation Eq. (A6) becoes an Haronic oscillator potential. Finally the significative values of r stay in the neighborhood of the iniu of the effective hyperradial potential where r Aa ² in ters of the r.. s. radius a. For bosons in ground state 3 A / and particles stay inside the range of the potential. / r 3 a does not vary uch when all For ferions f leads to 1 3 ( ) A 4/3 1/6 /.8458 /. Assuing a constant nuclear density r A r decreasing slowly for increasing A. 3 /3 a² r A for r 1. 5 Acknowledgeent I would like to thank S. Y. Larsen for help and advices in the writing of the paper. Appendix (see [3]) One starts fro the transforation (A1) P( z, r) g ( x, r) / g( x, r ) with z x² / r² 1, x r, (A) g ( x, r) N (1 x² / r²) xw[] ( z ) (note the isprint in ref [5] Eq. (3) where z ust be substituted for z ² ) where W []( z ) x (1 x ² / r ²), ( D 5) / to get the equation in the variable x : (A3) " d² g x² ( V ( x, r) U ( r)) / (1 ) g ( x, r) dx² g ² r² g ( x, r) V x r A A I x r ² 1 x² / r² (, ) ( ( 1) / 1) (, ) with V ( x, r) V ( x) V ( r ) r g ( x', r) g( x', r) (A4) I 1 x'² / r² [] [] [] (1 ak PK ( x) PK ( x')) dx '

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