Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics (2508.19022v1)

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 SUSY 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.