Weighted-residual methods for the solution of two-particle Lippmann-Schwinger equation without partial-wave decomposition

TL;DR

This paper develops weighted-residual computational schemes, including Galerkin, collocation, and Schwinger variational methods, to solve two-particle Lippmann-Schwinger equations without partial-wave decomposition, demonstrating improved convergence and a new hybrid approach.

Contribution

It introduces a unified weighted-residual framework with local basis functions for solving multi-variable integral equations without partial-wave expansion, including a novel hybrid-collocation method.

Findings

Schwinger variational method shows better convergence than Galerkin and collocation.

The hybrid-collocation method yields promising results.

The approach avoids partial-wave decomposition, simplifying quantum scattering calculations.

Abstract

Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves.The aim of the present work is to show that the weighted-residual approach in combination with local basis functions give rise to convenient computational schemes for the solution of the multi-variable integral equations without the partial wave expansion.The weighted-residual approach provides a unifying framework for various variational and degenerate-kernel methods for integral equations of scattering theory. Using a direct-product basis of localized quadratic interpolation polynomials,Galerkin, collocation and Schwinger variational realizations of the weighted-residual approach have been implemented for a model potential. It is demonstrated that, for a given expansion basis, Schwinger variational method exhibits better convergence with basis size than Galerkin and collocation methods. A novel hybrid-collocation method is implemented with promising results as well.