Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics

Abstract: The Wronskian formulation of supersymmetric quantum mechanics (SUSYQM) confluent transformation pairs is applied to the construction of phase-equivalent potentials with different bound spectra, replacing integral formulas. This allows to unify the two steps of a SUSYQM inversion scheme consisting in (i) the construction of a unique bound-state-less potential, possibly singular, from phase-shift inversion by a chain of non-confluent SUSYQM transformations, and (ii) the phase-equivalent addition of bound states by confluent SUSYQM pairs. Both steps are now combined in a single Wronskian formula, providing an elegant complete solution to the fixedangular-momentum inversion problem. This formalism is applied to the inversion of 3S1 and 1S0 neutron-proton data and its numerical implementation is discussed.

• The paper introduces a unified Wronskian formula that merges non-confluent and confluent SUSY transformations into a single-step inversion framework.
• It reconstructs phase-equivalent potentials with controlled bound-state spectra, demonstrated in both triplet and singlet neutron-proton scattering channels.
• Analytical tractability is achieved for moderate orders while numerical challenges for high-order Wronskians highlight the need for robust evaluation methods.

Unified Wronskian Formulation of Inverse Scattering with Supersymmetric Quantum Mechanics

Introduction and Context

The paper presents a unified analytical framework for the quantum inverse scattering problem at fixed angular momentum, leveraging the formalism of supersymmetric quantum mechanics (SUSYQM) and recent advances in confluent @@@@2@@@@ transformations. The central objective is to reconstruct interaction potentials from experimental phase shifts and bound-state data, a problem traditionally addressed via the Gel'fand-Levitan and Marchenko integral equations. The SUSYQM-based inversion method, in contrast, offers an algebraic approach, enabling the construction of phase-equivalent potentials with controlled bound-state spectra.

Historically, the SUSYQM inversion scheme has been divided into two steps: (1) construction of a unique, bound-state-less (possibly singular) potential via non-confluent SUSY transformations, and (2) phase-equivalent addition of bound states using confluent SUSY pairs. The main contribution of this work is the unification of these steps into a single Wronskian formula, thus providing a complete and elegant solution to the fixed-angular-momentum inverse problem.

Theoretical Framework

SUSYQM Transformations and the Inverse Problem

The starting point is the radial Schrödinger equation for a two-body system, with the potential $V_0(r)$ and singularity parameter $n_0$. SUSYQM transformations are constructed using factorization solutions at chosen energies (factorization energies), leading to new potentials $V_1(r)$ via the standard Darboux transformation:

$$V_1(r) = V_0(r) - 2 \frac{d^2}{dr^2} \ln \varphi_0(r, \mathcal{E})$$

where $\varphi_0$ is a solution at energy $\mathcal{E}$. Chains of such transformations at distinct energies (non-confluent) yield the Crum-Krein formula for the $M$-th transformed potential:

$$V_M(r) = V_0(r) - 2 \frac{d^2}{dr^2} \ln W[\varphi_0(r, \mathcal{E}_1), ..., \varphi_0(r, \mathcal{E}_M)]$$

This step allows fitting the scattering phase shift but results in a potential without bound states, and possibly with singular behavior at the origin.

Phase-Equivalent Transformations and Bound-State Engineering

To incorporate bound states without altering the phase shift, confluent SUSY transformations are employed. These involve pairs of transformations at the same energy, leading to phase-equivalent potentials. The addition (or removal) of bound states is controlled by the choice of factorization solutions and an arbitrary parameter $\beta$ that tunes the asymptotic normalization constant (ANC) of the bound state.

Previously, the final potential after bound-state addition was only available in integral form, which precluded a unified analytical treatment. The key advance in this work is the application of the Wronskian formulation for multi-confluent chains, as developed in recent literature, to obtain a closed-form expression for the final potential.

Unified Wronskian Formula

The main result is the following Wronskian formula for the potential after $M$ non-confluent and one confluent SUSY pair (i.e., addition of a bound state):

$$V_{M+2}(r) = V_0(r) - 2 \frac{d^2}{dr^2} \ln W\left[\varphi_0(r, \mathcal{E}_1), ..., \varphi_0(r, \mathcal{E}_M), u_0(r, \mathcal{E}_{M+1}), u_1(r, \mathcal{E}_{M+1})\right]$$

where $u_0$ is a solution at the bound-state energy and $u_1$ is a generalized eigenfunction constructed as:

$$u_1(r, \mathcal{E}_{M+1}) = \beta \varphi_0^\perp(r, \mathcal{E}_{M+1}) + \frac{\partial}{\partial \mathcal{E}} \varphi_0(r, \mathcal{E}) \bigg|_{\mathcal{E} = \mathcal{E}_{M+1}}$$

This formula expresses the final potential entirely in terms of solutions of the original potential $V_0$, enabling analytic construction when $V_0$ is simple (e.g., zero potential).

Applications to Neutron-Proton Scattering

$^3S_1$ (Triplet) Channel

For the triplet neutron-proton channel, the method is applied to construct potentials reproducing the low-energy phase shift and supporting a single bound state (the deuteron). Starting from $V_0 = 0$, two non-confluent transformations fit the phase shift, and a confluent pair adds the bound state. The resulting potential is:

$$V_4(r) = -2 \frac{d^2}{dr^2} \ln W\left[\sinh(\kappa_0 r), \sinh(\kappa_1 r), e^{-\kappa_2 r}, \beta e^{\kappa_2 r} + r e^{-\kappa_2 r}\right]$$

where $\kappa_0, \kappa_1$ are fixed by the phase shift, and $\kappa_2, \beta$ control the bound-state energy and ANC, respectively. This provides a family of phase-equivalent potentials, with the physical deuteron potential corresponding to specific parameter choices.

$^1S_0$ (Singlet) Channel

For the singlet channel, six non-confluent transformations fit the entire elastic phase shift, yielding a potential with no physical bound state. The addition of a "forbidden" bound state (as in Moscow-type deep potentials) is achieved via a confluent pair, resulting in:

$$V_8(r) = -2 \frac{d^2}{dr^2} \ln W\left[\sinh(\kappa_0 r), ..., \sinh(\kappa_3 r), e^{-\kappa_4 r}, e^{-\kappa_5 r}, e^{-\kappa_6 r}, \beta e^{\kappa_6 r} + r e^{-\kappa_6 r}\right]$$

This construction enables systematic exploration of deep potentials with tunable forbidden states, relevant for modeling Pauli effects in nucleon-nucleon interactions.

Numerical Implementation and Limitations

The unified Wronskian formula is analytically tractable for moderate numbers of transformations and simple $V_0$. For the $^3S_1$ channel, numerical evaluation is straightforward and confirms the expected phase-shift and bound-state properties. However, for the $^1S_0$ channel with a high-order Wronskian, symbolic and numerical evaluation becomes unstable due to the large determinant size and numerical cancellations. This highlights a practical limitation: while the formalism is general, robust numerical algorithms for high-order Wronskians are required for applications involving many transformations.

Implications and Future Directions

The unified Wronskian formulation provides a powerful analytical tool for constructing phase-equivalent potentials with arbitrary bound-state content, directly from experimental data. This has several implications:

• Analytical Control: The method allows explicit parameterization of the bound-state spectrum and ANC, facilitating systematic studies of phase-equivalent families and their physical properties.
• Inverse Problem Solving: The approach offers an alternative to integral-equation methods, with potential advantages in transparency and computational efficiency for low to moderate transformation orders.
• Nuclear Physics Applications: The construction of deep potentials with forbidden states opens avenues for improved modeling of nucleon-nucleon interactions, including the possibility of reducing partial-wave dependence by exploiting bound-state degrees of freedom.
• Algorithmic Challenges: The instability of high-order Wronskian evaluation suggests the need for new numerical techniques, possibly leveraging recurrence relations or specialized determinant algorithms.

Conclusion

The paper establishes a unified Wronskian framework for the SUSYQM-based inverse scattering problem, enabling analytic construction of phase-equivalent potentials with prescribed bound-state spectra. The formalism is demonstrated on neutron-proton scattering, yielding compact analytical potentials for both physical and deep (forbidden-state) cases. While the approach is limited by numerical challenges for large transformation chains, it provides a significant advance in the analytic treatment of the inverse quantum scattering problem and sets the stage for further developments in both mathematical physics and nuclear phenomenology.