Weighted-Residual Methods for the Solution of Two-Particle Lippmann–Schwinger Equation Without Partial-Wave Decomposition

2016

Phys. Rev. E

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Direct numerical solution of the coordinate-space integral-equation version of the two-particle Lippmann Schwinger (LS) equation is considered as a means of avoiding the shortcomings of partial-wave expansion at high energies and in the context of few-body problems. Upon the regularization of the singular kernel of the three-dimensional LS equation by a subtraction technique, a three-variate quadrature rule is used to solve the resulting nonsingular integral equation. To avoid the computational burden of discretizing three variables, advantage is taken of the fact that, for central potentials, azimuthal angle can be integrated out leaving a two-variable reduced integral equation. Although the singularity in the the kernel of the two-variable integral equation is weaker than that of the three-dimensional equation, it nevertheless requires careful handling for quadrature discretization to be applicable. A regularization method for the kernel of the two-variable integral equation is derived from the treatment of the singularity in the three-dimensional equation. A quadrature rule constructed as the direct-product of single-variable quadrature rules for radial distance and polar angle is used to discretize the two-variable integral equation. These twoand three-variable methods are tested on a model nucleon-nucleon potential. The results show that Nystrom method for the coordinate-space LS equation compares favorably in terms of its ease of implementation and effectiveness with the Nystrom method for the momentum-space version of the LS equation .

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2016

Phys. Rev. E

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“…Therefore, there is considerable interest in multi-variable methods that could produce the three-dimensional off-shell T-matrix in more economical manner than the Nystrom method. Galerkin [2,4], collocation [10], and Schwinger variational methods [10] have been investigated with various choices of multi-variable bases. These methods can effectively be viewed as contractions of the linear system of equations that the Nystrom approach gives rise to.…”

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confidence: 99%

“…These methods can effectively be viewed as contractions of the linear system of equations that the Nystrom approach gives rise to. In other words, the (large) equation system of the Nystrom method is replaced by a smaller set of approximate equations, by demanding that a residual vanishes on a chosen test space [10].…”

mentioning

confidence: 99%

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

2014

J Math Chem

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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

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

2014

Few-Body Syst

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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