The Petrov–Galerkin approach as a unifying framework for numerical solutions of the Lippmann–Schwinger equation

Abstract This paper aims to establish the Petrov–Galerkin approach as a unifying framework for various variational and degenerate kernel methods for the solution of the Lippmann–Schwinger (LS) equation of quantum scattering theory. Both transition-operator and resolvent-operator versions of the LS equation in momentum representation are considered. Within the PG formulation, commonly used schemes such as Galerkin, collocation, Schwinger variational and Bateman methods are shown to arise as specific realizations corresponding to particular choices of the approximation and test spaces. Interconnections of different realizations of the PG approach with finite-rank approximations of potential and free resolvent operators via oblique projectors are explored. In particular, new variants of Schwinger and Newton variational methods are obtained as special realizations of the PG approach. A total of sixteen distinct realizations of the Petrov–Galerkin and projection schemes are tested on a model s-wave scattering problem, and representative numerical results are presented to benchmark and compare the various formulations.