Gherghetta and Nambu (1993) constructed an exact nonlocal separable potential that reproduces a given ss-wave phase shift with an arbitrary number of zeroes of sin⁡δ0\sin\delta_0, applied it to 4^4He–4^4He scattering with its weakly bound dimer, benchmarked the off-shell continuation only against a solvable square-well surrogate, and left the many-body application to liquid 4^4He as an outlook. Four issues were left open: the construction was never carried beyond the ss wave; the off-shell behavior was never compared with a realistic local 4^4He potential; the free parameters had to be adjusted energy by energy, so no fixed potential was defined; and the claimed diagrammatic simplification of the many-body problem was never demonstrated. No subsequent work is known to extend, validate, or apply this construction, although adjacent methods (EST-type separable expansions, wave-function-informed separable fits for 4^4He, interval-based inverse constructions for nucleon–nucleon scattering) achieve parts of the underlying goals by other routes. The problem is to complete and validate the construction for 4^4He — the partial-wave set fixed in requirement (1), verified on-shell reproduction, and quantified off-shell fidelity against a realistic local reference including a principled treatment of the residual off-shell ambiguity; the many-body application to liquid 4^4He, discussed by the source only as an outlook, remains an optional direction beyond the completion boundary.

Background

Scattering data and the inverse problem. In nonrelativistic two-body quantum scattering, each partial wave (angular momentum ll; the ss wave is l=0l=0) is characterized on shell by a phase shift δl(k)\delta_l(k), where kk is the relative momentum: the on-shell scattering amplitude of that partial wave is eiδlsin⁡δl/ke^{i\delta_l}\sin\delta_l/k. The full two-body transition matrix (t-matrix) T(k′,k;E)T(k',k;E), evaluated at off-shell momenta k′≠kk'\neq k and general energies EE, is what actually enters three-body and many-body calculations, but it is not directly observable. The inverse scattering problem asks for an interaction potential that reproduces given scattering data — for local potentials this is the classical Gel'fand–Levitan–Marchenko theory, while for special potential classes one can write the inversion down explicitly.

Separable nonlocal potentials. A rank-one separable potential has the momentum-space form V(k,k′)=λ u(k)u(k′)V(k,k')=\lambda\,u(k)u(k') with a form factor uu; finite sums of such terms give finite-rank potentials. Because the kernel factorizes, the Lippmann–Schwinger equation (the integral equation for the t-matrix) collapses to algebraic equations, so the amplitude, bound states, and off-shell continuation all come out in closed form. Yamaguchi (1954) introduced the prototype; Gourdin and Martin (1958) and Tabakin (1969) posed and solved the inverse problem for such potentials — reconstructing uu and λ\lambda from a given phase shift rather than fitting them.

The zero-counting obstruction and the 1993 construction. By Levinson's theorem the net phase advance satisfies δ0(0)−δ0(∞)=Nπ\delta_0(0)-\delta_0(\infty)=N\pi (with NN the number of ss-wave bound states), and the Gourdin–Martin solution assumed that sin⁡δ0\sin\delta_0 has at most NN zeroes. Realistic short-range-repulsive (hard-core-like) interactions, however, drive the phase shift through many multiples of π\pi, producing more zeroes of sin⁡δ0\sin\delta_0 than NN; the atom–atom 4^4He interaction is a textbook case, combining a stiff repulsive core with a shallow attractive well that supports a dimer bound only by about one millikelvin. Gherghetta and Nambu (1993) extended the separable inverse solution to this case: writing x=k2x=k^2, they partition the energy axis into intervals (xn,xn+1)(x_n,x_{n+1}) between successive zeroes of sin⁡δ0\sin\delta_0, and on each interval define a partial phase (Omnes-type) factor eαn,n+1(x)e^{\alpha_{n,n+1}(x)} with

αn,n+1(x)=−1π∫xnxn+1δ0(x′)x′−x−iϵ dx′,\alpha_{n,n+1}(x)=-\frac{1}{\pi}\int_{x_n}^{x_{n+1}}\frac{\delta_0(\sqrt{x'})}{x'-x-i\epsilon}\,dx',

a Hilbert transform of the phase shift over the interval. Endpoint residues of the analytically continued integral equation are absorbed into an energy-dependent normalization factor F(E)F(E), and the full potential is a sum of intervalwise separable terms with rational functions P(x)/Q(x)P(x)/Q(x) carrying free constants. The authors constructed the 4^4He solution from the numerically computed phase shift of the Aziz potential (the standard semi-empirical local pair potential for helium, Aziz et al. 1979), incorporated the weakly bound state through the choice of P/QP/Q, compared the off-shell ratio u~(k′)/u~(k)\tilde u(k')/\tilde u(k) against an exactly solvable local square well chosen to mimic 4^4He, and closed with a discussion of how the factorized interaction

U(r,r′)=−ℏ22μλ u(r)u(r′)U(r,r')=-\frac{\hbar^2}{2\mu}\lambda\,u(r)u(r')

might simplify the many-body theory of liquid 4^4He by decoupling relative and center-of-mass coordinates, reportedly reducing a tt-channel diagram summation to ss-channel annihilation graphs (the channel labels classify two-line Feynman graphs by how the external legs are paired: tt-channel graphs exchange the incoming particles between the two lines, ss-channel graphs annihilate them into a common intermediate state).

What was left open. (i) The construction is worked out only for the ss wave; the authors state that analogous formulas follow for arbitrary angular momentum via a Hankel transform (a partial-wave projection through spherical Bessel functions), but do not carry this through. (ii) The off-shell benchmark is a square-well surrogate whose bound state is orders of magnitude deeper than the physical dimer, not the true local 4^4He interaction, so the off-shell fidelity of the reconstructed kernel is unknown. (iii) The free constants in P/QP/Q are not unique and in some energy regions must be retuned at each on-shell energy, so no fixed potential is defined; this reflects the structural fact that on-shell data alone cannot fix the off-shell continuation — phase-equivalent nonlocal potentials form infinite families (Muzafarov 1988). (iv) The many-body usefulness is asserted as an outlook only. Neighboring capabilities exist by other routes: the Ernst–Shakin–Thaler (EST) procedure (Ernst, Shakin, Thaler 1973) builds finite-rank separable potentials that reproduce the half-off-shell t-matrix of any chosen local potential at selected energies in any partial wave, and separable approximations of this type were applied to the helium trimer (Loucks and Levinger 1991); but none of these validates the phase-shift-inversion construction itself.

Problem Statement

Complete and validate the nonlocal separable inverse-scattering construction of Gherghetta and Nambu for the 4^4He–4^4He system, as one linked objective: obtain a usable, validated nonlocal separable representation of the 4^4He–4^4He interaction derived by inverse scattering from on-shell data. Concretely, the requested results are: (1) carry out the stated generalization of the intervalwise construction to arbitrary angular momentum — derive the partial-wave form factors — and exhibit the construction for the partial waves that contribute appreciably to low-energy 4^4He–4^4He scattering — for these identical bosons, the even partial waves l=0,2,4,…l=0,2,4,\ldots (enumerate the set carried out and fix lmax⁡l_{\max} by a stated inclusion criterion, e.g. a phase-shift or cross-section threshold, requiring at least l=0l=0 and l=2l=2), with the energy range used stated explicitly; (2) verify the construction on shell by solving the two-body scattering problem with the resulting multi-rank separable potential and confirming that it reproduces the input phase shifts in each partial wave in which the construction is carried out, together with the weakly bound dimer state; and (3) quantify off-shell fidelity by replacing the square-well surrogate with a realistic local 4^4He reference potential — the reference being taken as the Aziz et al. (1979) potential from which the source construction derived its input phase shifts: compute the reference half-off-shell t-matrix (the t-matrix with one of its two momentum arguments fixed on shell), measure the deviation of the separable kernel's off-shell ratios over the momentum range in which the construction is carried out, and identify criteria — or the additional off-shell input — that select an adequate continuation, recognizing that on-shell data alone cannot fix one uniquely. The many-body application to liquid 4^4He appears in the source only as an outlook (a discussion of how the factorized form might simplify the many-body problem); a controlled many-body demonstration with the constructed potential is a possible extension beyond the completion boundary and is not required to resolve this problem. The completion boundary is: a fixed parameter assignment that reproduces the on-shell data in the partial-wave set fixed in requirement (1) — any residual freedom frozen by an explicitly stated rule; and a quantified statement of off-shell deviation from the local reference together with the criterion (or the additional off-shell input) that selects the continuation being reported.

References

  1. T. Gherghetta and Y. Nambu, "Nonlocal separable solutions of the inverse scattering problem", International Journal of Modern Physics A 8, 3163–3184 (1993). DOI: 10.1142/S0217751X93001260, https://doi.org/10.1142/S0217751X93001260.
  2. Yoshio Yamaguchi, "Two-Nucleon Problem When the Potential Is Nonlocal but Separable. I", Physical Review 95, 1628–1634 (1954). DOI: 10.1103/PhysRev.95.1628, https://doi.org/10.1103/PhysRev.95.1628.
  3. M. Gourdin and A. Martin, "Exact determination of a phenomenological separable interaction", Il Nuovo Cimento 8, 699–707 (1958). DOI: 10.1007/BF02815249, https://doi.org/10.1007/BF02815249.
  4. Frank Tabakin, "Inverse Scattering Problem for Separable Potentials", Physical Review 177, 1443–1451 (1969). DOI: 10.1103/PhysRev.177.1443, https://doi.org/10.1103/PhysRev.177.1443.
  5. R. A. Aziz, V. P. S. Nain, J. S. Carley, W. L. Taylor, and G. T. McConville, "An accurate intermolecular potential for helium", The Journal of Chemical Physics 70, 4330–4342 (1979). DOI: 10.1063/1.438007, https://doi.org/10.1063/1.438007.
  6. D. J. Ernst, C. M. Shakin, and R. M. Thaler, "Separable Representations of Two-Body Interactions", Physical Review C 8, 46–52 (1973). DOI: 10.1103/PhysRevC.8.46, https://doi.org/10.1103/PhysRevC.8.46.
  7. V. M. Muzafarov, "Inverse scattering problem for a family of phase-equivalent nonlocal potentials", Inverse Problems 4, 185–209 (1988). DOI: 10.1088/0266-5611/4/1/016, https://doi.org/10.1088/0266-5611/4/1/016.
  8. Roger Loucks and J. S. Levinger, "Separable Approximations to the Helium Trimer", Annalen der Physik 503, 115–120 (1991). DOI: 10.1002/andp.19915030111, https://doi.org/10.1002/andp.19915030111.
  9. N. H. Kwong and H. S. Köhler, "Separable NN potentials from inverse scattering for nuclear matter studies", Physical Review C 55, 1650–1664 (1997). DOI: 10.1103/PhysRevC.55.1650, https://doi.org/10.1103/PhysRevC.55.1650.
  10. Carl M. Bender and Hugh F. Jones, "Bound states of PT-symmetric separable potentials", Physical Review A 84, 032103 (2011). DOI: 10.1103/PhysRevA.84.032103, arXiv:1107.2293, https://doi.org/10.1103/PhysRevA.84.032103.
  11. C. González-Santander and F. Domínguez-Adame, "Non-local separable solutions of two interacting particles in a harmonic trap", Physics Letters A 375, 314–317 (2011). DOI: 10.1016/j.physleta.2010.12.013, https://doi.org/10.1016/j.physleta.2010.12.013.
  12. Jing-Lun Li and Shu-Lin Cong, "High-rank separable atom-atom interaction potential used for solving two-body Lippmann-Schwinger and three-body Faddeev equations", The Journal of Chemical Physics 149, 204109 (2018). DOI: 10.1063/1.5045084, https://doi.org/10.1063/1.5045084.
  13. H. F. Arellano and N. A. Adriazola, "Inverse-scattering separable NN potential constrained to phase-shift data up to 2.5 GeV. I.—Uncoupled states", The European Physical Journal A 60, 158 (2024). DOI: 10.1140/epja/s10050-024-01367-7, arXiv:2406.12827, https://doi.org/10.1140/epja/s10050-024-01367-7.