[Paper Review] Momentum-space Lippmann-Schwinger-Equation, Fourier-transform with Gauss-Expansion-Method

Th. A. Rijken|arXiv (Cornell University)|Sep 19, 2014

Experimental and Theoretical Physics Studies5 references4 citations

TL;DR

This paper presents a novel method to compute momentum-space nucleon-nucleon potentials from configuration-space potentials using the Gaussian-Expansion-Method (GEM) for Fourier transformation. By expressing the potential in Gaussian basis functions with mass parameters set via the Hiyama-Kamimura geometric progression recipe, the method enables analytical evaluation of Fourier-Bessel integrals, eliminating numerical oscillations—especially for unequal initial and final momenta. The approach yields momentum-space potentials that reproduce NN phase shifts with high accuracy, matching configuration-space results for the ESC08c potential.

ABSTRACT

In these notes we construct the momentum-space potentials from configuration-space using for the Fourier-transformation the Gaussian-Expansion-Method (GEM). This has the advantage that the Fourier-Bessel integrals can be performed analytically, avoiding possible problems with the oscillations in the Bessel functions for large r, in particular for $p_f eq p_i$. The mass parameters in the exponentials of the Gaussian base-functions are fixed using the geometric progression recipe of Hiyama-Kamimura. The fitting of the expansion coefficients is linearly and very fast. Application to nucleon-nucleon is given in detail for the recent Extended-soft-core model ESC08c. The NN phase shifts obtained by solving the Lippmann-Schwinger equations agree very well with those obtained in configuration-space solving the Schrödinger equations.

Motivation & Objective

• To develop a robust, analytical method for transforming configuration-space nucleon-nucleon potentials into momentum-space form.
• To overcome numerical instabilities in Fourier-Bessel integrals arising from oscillatory Bessel functions, particularly when initial and final momenta differ.
• To enable efficient and accurate solution of the Lippmann-Schwinger equation in momentum space using analytically derived potentials.
• To generalize the method to coupled-channel hyperon-nucleon systems by extending expansion coefficients to matrix form.

Proposed method

• The Gaussian-Expansion-Method (GEM) is used to expand configuration-space potentials in a basis of Gaussian functions with fixed mass parameters via the Hiyama-Kamimura geometric progression recipe.
• Fourier-Bessel integrals for central, tensor, spin-orbit, and quadratic-spin-orbit potentials are evaluated analytically using known integral forms involving error functions and modified Bessel functions.
• The method avoids numerical integration of oscillatory Bessel functions by leveraging the analyticity of Gauss-Bessel integrals, ensuring stability and precision.
• Expansion coefficients are determined via linear fitting to configuration-space potential values and volume integrals, enabling fast and accurate reconstruction of momentum-space potentials.
• The approach is applied to the extended soft-core ESC08c potential, with results validated by solving the Lippmann-Schwinger equation using Kowalski-Noyes and Haftel-Tabakin methods.
• For coupled channels (e.g., ΛN, ΣN), the expansion coefficients are generalized to 2×2 matrices in isospin and channel space.

Experimental results

Research questions

• RQ1 Can the Fourier transformation of configuration-space potentials into momentum space be performed analytically to avoid numerical oscillations in Bessel functions?
• RQ2 Does the Gaussian-Expansion-Method with Hiyama-Kamimura mass parameter selection yield momentum-space potentials that reproduce phase shifts accurately?
• RQ3 Can the method be generalized to coupled-channel hyperon-nucleon systems with matrix-valued expansion coefficients?
• RQ4 How does the computational efficiency of the GEM-based momentum-space potential compare to direct numerical integration in momentum space?

Key findings

• The analytical evaluation of Fourier-Bessel integrals using the GEM ensures numerical stability and eliminates oscillations, particularly for p_f ≠ p_i.
• The momentum-space potentials constructed via GEM reproduce NN phase shifts from the ESC08c potential with excellent agreement to those obtained in configuration space.
• The method achieves high accuracy with only a few GEM basis functions, as the fitting of expansion coefficients is linear and converges rapidly.
• The use of the Schwinger representation allows a natural derivation of the Gaussian form, linking meson propagators to the GEM basis functions.
• The Kowalski-Noyes and Haftel-Tabakin methods for solving the Lippmann-Schwinger equation yield essentially identical phase shifts, validating the method's consistency.
• The approach is readily generalizable to coupled-channel systems, such as hyperon-nucleon scattering, by extending expansion coefficients to matrix form in channel space.