Padé Phenomenology for Two-Body Bound States Padé Phenomenology for Two-Body Bound States

Pade approximants in the squared momentum variable, recently used for elastic scattering, are employed in generating accurate analytic approximants for bound states. Through iteration, [L/L+1] approximants yield the lowest eigenstate of the homogeneous Lippmann-Schwinger equation for Yukawa, Malfliet-Tjon, and Reid soft core central potentials with, respectively, L =1, 2, and 3. Higher eigenstates are readily obtained; the second is given for the Yukawa potential. Analytic separable expansions and scattering expressions result. NUCLEAR STRUCTURE Fade approximants in k, analytic two-body bound states, separable expansions, effective range parameters.

(Received 9 August 1982) Pade approximants in the squared momentum variable, recently used for elastic scatter- ing, are employed in generating accurate analytic approximants for bound states.Through iteration, [L/L+1] approximants yield the lowest eigenstate of the homogeneous Lippmann-Schwinger equation for Yukawa, Malfliet-Tjon, and Reid soft core central po- tentials with, respectively, L =1, 2, and 3. Higher eigenstates are readily obtained; the second is given for the Yukawa potential.Analytic separable expansions and scattering ex- pressions result.
NUCLEAR STRUCTURE Fade approximants in k, analytic two- body bound states, separable expansions, effective range parameters.Two-body bound states, most notably the deu- teron, have long been the subject of analytic investi- gations.However, analytic methods have met with limited success.
Although the Rayleigh-Ritz method has proved useful, ' as have iterative techniques, the former leaves unsolved the important problem of basis optimization, while the latter can lead to rapid growth of complexity with each itera- tion.Even though in the computer era the two- body problem no longer demands analytic solutions, they are patently desirable because of their tractabil- ity and their possible use as stepping stones to solu- tions of few-body problems.For example, their use in constructing separable representations of potentials can have a decisive advantage in three ' and four body problems.In this paper we introduce a new method of using Pade approximants (PA) (Ref.7) for iteratively generating analytic solutions of two-body bound-state problems that is accurate and simple to use, and we use this method to construct analytic separable expansions and expressions for effective range parameters.
A PA is defined as a rational function approxi- mant, with its coefficients determined by equating the truncated Taylor expansions of the PA and the function being approximated.
Previous work has shown reasonable analytic approximants in momen- tum space to possess a simple pole structure such as naturally arises from using PA.Also, it has re- cently been found that x -=k is a natural variable for PA to the two-body scattering function, F(x)=k cot(5o), associated with realistic NN poten- tials, and that such approximants closely reproduce exact results.The bound state.results we give here V(r)= g VJ PJP The two-body Schrodinger equation for a bound S state, ' -1 k P(k)= E-M f, dy y'Vo«y)4(y) from using PA also yield rational scattering func- tions when employed with the unitary pole approxi- mation (UPA) (Ref.9) or the unitary pole expansion (UPE).'o When applicable, the present method ap- pears to be a powerful alternative to previous uses of PA's for solving the integral equations of few- body systems.
The present method consists of assuming a Pade form for the solution of a two-body bound state problem and recovering that same form from a series expansion of the integral equation.Unlike the Thomas-Sachs method, ours leaves the complexity of our eigenfunction unchanged during the iteration process.We consider the case where the potential is a sum of Yukawas; for numerical results we further specialize to a single Yukawa which models the triplet np interaction (Y), the 'So Malfhet-Tjon potential (MT), " which is a sum of two Yukawas, and the 'So Reid soft core potential (RSC), ' which is a sum of three Yukawas.The last two, especially the RSC, have strong short-range repulsions.
For Y, B=2.240 MeV; for MT and RSC, we set the binding energy 8 to 0, following

Harms. '
The potential is written: 1982 The American Physical Society when written with the potential of Eq. ( 1) in the momentum representation, becomes where we use natural units (fi=c=l), we define strengths gj --MV-J /pj (M/2 is the reduced mass) and yo MB --(B is the binding energy), and we have multiplied out the bound state pole: The overall strength parameter A, is introduced as an eigenvalue and equals unity if the potential produces the binding energy E = -B.Our equation for X(k} is identical to the homogeneous Lippmann-Schwinger equation used by Harms to generate the UPE, and X(k) becomes the form fac- tor for the UPA when A, =1: where XJ(k) are eigenfunctions of the Lippmann-Schwinger kernel with ME=yo and having eigenvalues Aj.Orthonormality holds in the form x" I G,( -B) I X.) = -s where Go(E) =(Ek /M) The solutions X"(k) and A,"are expressible, for E&0, in terms of an eigenvalue problem for a Her- mitian operator, Go VGO .Consequently, prop- erties of the spectrum are readily obtained and the Rayleigh-Ritz method produces variational bounds on eigenvalues.
We write the even function X(k) as a PA in the variable x =k: where m -1 is the number of poles of X(k) on the positive imaginary axis.The constants hq and dã re simply and analytically expressible in terms of the Pade coefficients and y]].A PA for X(k) is recovered and a value of A, obtained using the first 2L+2 terms of the series expansion of the right hand side of Eq. (5}.The solution, found iteratively by using the recovered X(k) as an input to the right hand side, generally requires about 30 iterations to get 14-figure convergence of all parameters.Table I gives our results for the three potential models, showing the lowest order PA required for conver- gence.In coordinate space the wave functions are sums of exponentials, all range parameters being uniquely determined by our method, while in momentum space the wave functions closely resem- ble those previously given.' ' The Yukawa model used by Harms' differs so slightly from the one we employ that comparisons to three figures can still be made.It is also possible to obtain bind- ing energies iteratively for those potentials that have bound states, by varying B until A, equals unity.For our Yukawa model, this yields B=2.202 MeV, pro- ducing the characteristically larger error in B (1. 7%) than in A, (0.27%%uo).
Higher eigenstates can also be obtained either iteratively or by diagonalization, as will be dis- cussed in more detail in a forthcoming paper.The natural basis for either approach is the I + 1 di- mensional subspace spanned by the [L /L +1] PA's   possessing precisely the same denominator polyno- mials.
Table I reflects two approximations, the first be- ing our basic ansatz that a PA form adequately represents the eigenfunction.
Another approxima- tion is made in the iteration procedure, by truncating the series expansion of the right hand side of Eq. ( 5).We have tested the ansatz by doing a Rayleigh-Ritz calculation with PA functions as a TABLE I. First eigenvalue of Lippmann-Schwinger equation obtained through iteration compared with accurate results.
where L and M are the degrees of the polynomials Pt. and Q]]t, and we choose M =L+1, which as- sures normalizability.
The Hulthen function ob- tained by Sachs has a [0/1] structure.Anticipating our result that the stable solutions have poles on the positive imaginary axis, we obtain the equation n m X(k)=A, g g hqtan, (5)   p=l q=1 Potential Y MT RSC Iterative Accurate basis and using numerical quadrature.For simplicity, we shall restrict all further detailed discussion to the Yukawa model in the [1/2] approximation.
Figure 1 shows the lowest eigenstate, Xi(k), and the orthonormal [1/2] function, X2(k).They closely resemble the first two eigenfunctions numerically obtained for the Harms-Yukawa model.' The Rayleigh-Ritz procedure is easily applied in the Xi(k}, X2(k) subspace.For this purpose 20-point Gauss-Legendre integration is sufficiently accurate to compute matrix elements.If we write iX&=~t iXi&+p iX2&, then solving the equations &Xi I Go( -8)~IX&=~&X I Go( -8) IX& i=1,2, with A = VGp( 8) andthe orthonormality condition leads to the upper bounds k» (1.000036 and A, 2(3.479074.Our A, 2 for the Harms and Adhikari Yukawas are the same to three figures, compared with Harms's accurate value of 3.2846. We obtain a small admixture P=0.606128)&10 of ~Xq& to the first eigenstate.Hence, Gp( -8)A is practically diagonal in the Pade-iterated basis, and the plots of the new approximate eigenfunctions are indistinguishable from those in Fig. 1.Varying 8, we obtain 8=2.2395 for A, = 1, which is in striking agreement, differing by 0.02 percent from the exact binding energy.These results strengthen the credi- bility of our approach for generating analytic solu- tions introduced in this paper.
With our ~Xi & and ~Xq &, analytic approximations to the rank-1 UPA and the rank-2 UPE are 0. 6-immediately obtained, and the scattering function F(k ) =k cot(5p) is easily found from the R matrix.
The R-matrix integrals can all be evaluated analytically and lead to rational forms for F(k ).For the present Yukawa potential, convergence of the UPE series contributions to E(k ) is known to be slow, although it is much faster for potentials with short-range repulsions.' %e limit our discussion here to the scattering length.We define the integral dy y'Xi'(y) I(y )=M P y2+y 2 By the normalization condition, I(yp )=1.The UPA scattering length is n.MXi (0) a= 2(I(0) -A, i} '   (6)   and if we set A, i --1 we guarantee that the T matrix will have the correct bound state pole, and we ob- tain 0=5.614 fm compared with the more precise value of 5.470 fm that we computed from the Schrodinger equation in coordinate space.The nu- merical result just given comes from the use of the iterative Xi(k) and X2(k).Scattering length calcula- tions reported here are all insensitive to choice of basis, whether iterative or Rayleigh-Ritz.
Our two-term UPE leads to a scattering length of the form ai+a2 -2aia2J(0) a= 1 -aia2J (0) where a» and a2 are the scattering lengths associated with the two separable terms in V and ir(M/2)Xi(k)X2(k) J(k') k2 =P J dyy Xi(y)X2(y)Gp The weakly attractive second term in V associated with Xq(k) drives the total scattering length to a value of -0.7 fm when Rayleigh-Ritz eigenvalues are used.If A, i and A, 2 are allowed to vary, a better fit to low energy scattering is achieved at the cost of shifting the bound state pole in the T matrix.If A, » --1 is used, the bound state pole is reproduced, but the second term in V must be strongly repressed (A, 2= = 110) to fit the scattering length to the quot- ed precision.
Although these results are approximations to a UPE analysis, the simple analytic structures permit extensions with great ease.Under current investiga- tion is the possibility of fitting the bound state pole and the scattering function F(k ) at low energies FIG. 1. Iterative eigenfunction Xi(k) in [I/2] approxi- mation (solid line) and +2(k) (dashed line) obtained by orthogonalizing to g~(k) while maintaining the same Pade denominator, Q2(k ).Both functions are normal- ized.