Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann–Schwinger Equation for Two-Particle Scattering

Abstract

Finite-rank expansions of the two-body resolvent operator are explored as a tool for calculating the full three-dimensional two-body T-matrix without invoking the partial-wave decomposition. The separable expansions of the full resolvent that follow from finite-rank approximations of the free resolvent are employed in the Low equation to calculate the T-matrix elements. The finite-rank expansions of the free resolvent are generated via projections onto certain finite-dimensional approximation subspaces. Types of operator approximations considered include one-sided projections (right or left projections), tensor-product (or outer) projection and inner projection. Boolean combination of projections is explored as a means of going beyond tensor-product projection. Two types of multivariate basis functions are employed to construct the finite-dimensional approximation spaces and their projectors: (i) Tensor-product bases built from univariate local piecewise polynomials, and (ii) multivariate radial functions. Various combinations of approximation schemes and expansion bases are applied to the nucleon-nucleon scattering employing a model two-nucleon potential. The inner-projection approximation to the free resolvent is found to exhibit the best convergence with respect to the basis size. Our calculations indicate that radial function bases are very promising in the context of multivariable integral equations.