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. © 2014, Springer-Verlag Wien

Direct numerical solution of the Lippmann-Schwinger equation in coordinate space without partial-wave decomposition

“…However, more moderate values like N r = 176 and N x = 60 would be sufficient to obtain agreement within 5-6 significant figures. We note in passing that, reference results obtained from momentum-space Nystrom method [11,13] involves about the same level of computational effort as the coordinate-space Nystrom method for comparable levels of convergence .…”

“…[13], and are stable within seven digits after the decimal point to further variations in computational parameters . Tables 1 and 2 report results of Nystrom calculations with different values of N r , N x and N φ .…”

“…The radial basis function (rbf) approach (which is nearly dimension independent) has emerged during recent years as an alternative to direct-product bases [22,23]. In a recent article [13] we explored the use of rbf's (in momentum space)) in relation to momentum-space LS equation with promising results (for both three-dimensional and two-dimensional versions). A logical continuation of the present work would be to consider rbf expansions (in coordinate space) as a means of solving the coordinate-space LS equation and of obtaining separable expansions of the multivariable T-matrix.…”

“…Towards this end, multivariable methods has been investigated for the solution of the multi-variable LS equation in the momentum space [3][4][5][6][7][8][9][10][11][12][13]. For example in Refs.…”

“…Calculational schemes based on momentum-space LS equations dominate the literature for two-body scattering computations, as exemplified in [3,[5][6][7][8][9][10][11][12][13]. Coordinate-space version of the LS equation have received relatively less attention as a computational vehicle, although the coordinate-space partial-wave LS equation has been employed in connection with various types of Schwinger variational methods [16][17].…”