Rank-One Inverse Scattering Problem: Reformulation and Analytic Rank-One Inverse Scattering Problem: Reformulation and Analytic Solutions Solutions

Using the E-matrix formalism, we give a simplified reformulation of the S-wave rank-one inverse scattering problem. The resulting Cauchy integral equation, obtained differently by Gourdin and Martin in their first paper, is tailored to rational representations of F{k)=k cot{6O). Use of such F(k) permits a simple but general solution without integration, giving-analytic form factors having a pole structure like the S matrix that are reducible to rational expressions using Pade approxi-mants. Finally, we show a bound state pole condition is necessary, and makes the form factor unique.


Rank-One Inverse Scattering Problem: Reformulation and Analytic Solutions Rank-One Inverse Scattering Problem: Reformulation and Analytic Solutions
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This article is available at DigitalCommons@URI: https://digitalcommons.uri.edu/phys_facpubs/214 Such potentials are being developed to reproduce NN phase shifts with increasing accuracy. In particular, rank-two and rank-three interactions possessing form factors that are rational functions of momentum have recently been introduced.
It would be desirable for there to be a convergent procedure for constructing analytic low-rank potentials that (a) are interchangeable with mesontheoretic potentials in an appropriate energy range and (b) have a simple and direct connection with two-body data. Such a procedure would simplify realistic nuclear calculations, in some cases decisively, while helping to justify the use of finite rank representations, known not to be fully equivalent, at higher energies, to local potentials. ' In an approach which emphasizes the first goal, recent work based upon new Pade approximant methods ' for bound states can generate rational form factors for a variety of local potentials such as sums of Yukawas. Here we focus upon the second goal. We present a new formulation of rank-one inverse scattering theory that yields simple analytic form factors, and is designed to take advantage of the discovery that rational representations of the scattering function F(k)=k cot(5o) can be accurate. Relating to the first goal, we employ our formulation to construct a rank-one potential phase-shift equivalent to a Yukawa potential.
Despite an interesting attempt with a rank-one interaction, it is generally accepted that at least a rank of two is required to reproduce one salient feature of the NN interactions: a long range attraction together with a strong short range repulsion. ' A rank-two inverse scattering formalism, that of Fiedeldey, already exists. It allows for the initial introduction of a somewhat arbitrary long range attraction such as could be constructed using the formulation we present here. We will address the rank-two inverse scattering problem in another paper.
We present a E-matrix formulation of rank-one inverse scattering theory that accomplishes several things: simpler physical and mathematical analysis than previous formulations direct use of the known solutions of the inhomogeneous Riemann boundary value problem;" and presence of data in the form of k cot(5o), which provides a context for introducing Pade approximants, facilitates analytical continuation, and enables a pole analysis of form factors not previously carried out. When the scattering function F(k) is rational, we obtain the form factor without integrations. For a rational S matrix S(k) we find that the poles of S(k) and of the squared form factor h (k) coincide and are of the same order in the upper k plane, except at the bound state pole, ik~, and we obtain a general bound state pole condition: The order of the pole in h(ikii) must be one less than the order in S(ikii). This condition makes h (k) unique, and as seen in our example, is easily enforced.
Our work is aligned most closely with the initial approach of Gourdin  Although this expression can presumably be evaluated when F(k) is given in rational form, it would appear immensely more practical to understand the simpler approach we present and to avoid integrations altogether.
Subsequent work has led to the same expression for h (k).
Bolsterli and MacKenzie obtained this result starting with the T matrix, ' while Chadan and Sabatier explicitly referred to the Riemann problem while supplying extensions and more rigor to GMII. ' Our approach uses the same analyticity properties of scattering states but differs significantly in detail by working with a Cauchy integral equation.
Coupled with the high accuracy of rational scattering functions, our formulation provides a powerful tool for inverse scattering problems. We specialize to S-wave scattering states and an attractive interaction. The repulsive case is easily traced through upon changing the sign in Eq. (1). Given a scattering function F(k) =k cot(5p), we find the rank-one interaction producing F(k), if it exists, by (a) solving the S-wave K-matrix equation for a rank-one interaction, and (b) treating the formal solution as an integral equation in where Cauchy principal values are understood, obtaining where we have defined h (k) to be an even function and J(co)= f dy -00~y Equation (5) can be put in the standard form of the dominant integral equation of the Cauchy type for which a general solution is known. " We write (6) with a(k)=kcot(5p), b(k)=ik, and f(k) becoming the constant f-= f(k)=, -f dyh(y) .
In turn, Eq. (6) is reducible to a special case of the inhomogeneous Riemann boundary value problem, which is to The fully on-shell K matrix is related to the scattering function by Finally, the form factor is given upon application of the well-known Plemlj formulas, " also used to establish the connection between the Riemann problem and our integral equation: As For nucleons, it is realistic to assume there is no zero energy resonance and~5(0) -5(oo) (m, and we shall make these assumptions here. Then g is twice the number of bound states. ' If X=O, the requirement that h(k)~0 as k~oo makes the solution unique. If 7=2, our method will be shown to give the unique form factor consistent with the binding energy.
The general algorithm for solving the Riemann problem is somewhat complicated, especially in the case X&0, requiring the introduction of ancillary functions. " When F(k) is rational, the algorithms become simple. As the basic application of our formulation, we specialize in what follows to a rational F(k). Then where polynomials c+,d+ (c,d ) have roots in the upper (lower) half plane. Then g=m+ n+, where m+-(n+ ) is the number of zeros of c+ (d+ ), and the number of bound states is half the difference between the number of poles and zeros of the S matrix in the upper half plane.
When poles and zeros are counted here, it is their multiplicities that are added.
The decomposition of 6 in Eq. (12) permits the use of find functions P+(z) and P (z) that are analytic, respectively, in the upper (D+) and lower (D ) half complex plane with limiting values on the real k axis, P -(k), that satisfy P+(k) = G (k)P (k) +t (k) .
The coefficient of the Riemann problem is a (k) b(k) a(k)+b(k) ' which is simply related to the S matrix by G (k) =Sp (k).
The free term t (k) of the Riemann problem is proportional to the scattering amplitude: We require P-(Oo)=0 to restrict to solutions h(k) that also vanish at infinity. Consequently, by the generalized Liouville theorem, R (z) can be an arbitrary polynomial of degree X -1, written Rz i(z), but with R &(z)=0. The P+ -(k) are found by taking the real limit of z, z~k +-, of those functions analytic in D+ and D For a detailed discussion of the solution both for 7=0 and X = 2, we make use of the function Furthermore, the zeros must be distributed in each of the functions c+ (k) and d+(k) symmetrically about the imaginary z axis.
We are now able to examine the symmetry and analytic structure of the solution h (k) as a consequence of properties of the S matrix, including well-defined behavior of S(k) at the poles and real zeros of the analytically continued rational solution h(z). If h(z) has a zero for real z=+k"then A+(k, )=A (k, ) and 5(k, )=0. The existence of a zero of 5(k), analyzed in detail in Ref. 7, can be associated with a strong short-range repulsion, and will not be dealt with here. Then h(k) has no zeros and no sign changes for real finite k, and if attractive, satisfies h (k) & 0. A solution with h (k) & 0 for finite k would correspond to a repulsive interaction, as seen by starting with a different sign in Eq. (1). The explicit solution is given from Eqs. (14) and (11): is written from Eq. (5) as S(k) =A+(k)/A (k) .
Since A+(k)=A (k), it follows that S(k)=1/S(k), while from unitarity, S(k)*=1/S(k). It follows by substitutions from Eq. (12) that we may require  (12) and (17) it is evident that h (k) has the same poles as those of S(k) in D+, and that these poles have the same order unless the remaining constant Ct in R i(z) is assigned the value that creates a common factor k +kii in the numerator. This bound state condition in practice is simple to apply.
Our first example is for a shape-independent F(k) consisting of the first two terms in the effective range expansion. ikz. The bound state pole condition, easily applied, lowers the order of the pole of h (k) at iktt to unity. The solution is still given by Eq. (18) with p=0. These simple results may be compared with those given in Ref. 10. As a final example we consider a Yukawa potential that fits the 'So np effective range parameters. ' Figure 2 shows our rational least squares fits of Ii (k) that we computed for this potential. The [4/3]  (k -k2i) +k22 with a i --0.0201873 fm ', a2 --0.000811496 fm a3 ---0.0175540, k i --0.0667608 fm ', k2i --0.585295 fm ', k22 --1.20083 fm '. Here, rational approximants to g(k) would be most easly obtained by least squares methods. ' We are indebted to Jill C. Bonner for useful comments, and we also wish to thank R. G. Newton for suggestions which have led to substantial improvements in the manuscript.