Multi-variate Bateman method for two-body scattering without partial-wave decomposition

Abstract

The use of Bateman method for solving the two-variable version of the twobody Lippmann-Schwinger equation without recourse to partial-wave decomposition is investigated. Bateman method is based on a special kind of interpolation of the momentum representation of the potential on a multi-variate grid. A suitable scheme for the generation of a multi-variate Cartesian grid is described. The method is tested on the Hartree potential for electron-hydrogen scattering in the static no-exchange approximation. Our results show that the Bateman method is capable of producing quite accurate solutions with relatively small number of grid points.Keywords Quantum scattering theory · Lippmann-Schwinger equation · Few-body collisions · Multi-variate interpolation and approximation · Bateman interpolation · Degenerate-kernel methods for integral equations · Nystrom method · Faddeev equations

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

“…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.…”

“…For example in Refs. [11,12] , we have considered multivariable implementations of Schwinger variational and Bateman methods for two-body LS equation in momentum space. Significant progress has also been reported on the formal and computational aspects of solving the three-particle momentum-space Faddeev equations directly as 5-variable problems without invoking angular momentum decomposition [2,14,15].…”

“…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].…”

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

“…Certain drawbacks of this strategy have been noted in recent years, especially for high-energy collisions and within the context of few-body problems. As a result, computational methods that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves have been explored recently by a number of groups [1][2][3][4][5][6][7][8][9][10][11][12]. Various direct multivariable methods have been investigated for the solution of two-body Lippmann Schwinger (LS) equation.…”

“…Among these, multivariable version of the Schwinger variational method (with local interpolation polynomials as the expansion basis) and some of its variants have been shown to be quite effective [11]. In a similar vein, a multivariate Bateman interpolation of the momentum-space kernel of the potential have proved to be a relatively simple and effective method [12]. It is interesting to note that both these methods can also be viewed as the result of certain finite-rank separable approximations of the interaction potential [11,14].…”