Direct numerical solution of the two-particle Lippmann-Schwinger equation in coordinate space using the multi-variable Nystrom method

TL;DR

This paper develops a direct numerical method using the multi-variable Nystrom approach to solve the two-particle Lippmann-Schwinger equation in coordinate space, effectively handling singularities and reducing computational complexity for central potentials.

Contribution

It introduces a regularization and quadrature scheme for the two-variable integral equation, improving numerical solutions of the LS equation in coordinate space for central potentials.

Findings

The method effectively handles singular kernels in coordinate space.

Results compare favorably with momentum-space solutions.

The approach simplifies implementation and reduces computational load.

Abstract

Direct numerical solution of the coordinate-space integral-equation version of the two-particle Lippmann Schwinger (LS) equation is considered as a means of avoiding the shortcomings of partial-wave expansion at high energies and in the context of few-body problems. Upon the regularization of the singular kernel of the three-dimensional LS equation by a subtraction technique, a three-variate quadrature rule is used to solve the resulting nonsingular integral equation. To avoid the computational burden of discretizing three variables, advantage is taken of the fact that, for central potentials, azimuthal angle can be integrated out leaving a two-variable reduced integral equation. Although the singularity in the the kernel of the two-variable integral equation is weaker than that of the three-dimensional equation, it nevertheless requires careful handling for quadrature discretization to be applicable. A regularization method for the kernel of the two-variable integral equation is derived from the treatment of the singularity in the three-dimensional equation. A quadrature rule constructed as the direct-product of single-variable quadrature rules for radial distance and polar angle is used to discretize the two-variable integral equation. These two- and three-variable methods are tested on a model nucleon-nucleon potential. The results show that Nystrom method for the coordinate-space LS equation compares favorably in terms of its ease of implementation and effectiveness with the Nystrom method for the momentum-space version of the LS equation .