Application of wavelets to singular integral scattering equations

Abstract

The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms is demonstrated by solving the two-body T-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems.

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

“…However, for non-central potentials, partial waves are coupled and advantages of partial wave expansion dispappear to a large extent. Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves [1][2][3][4][5][6][7][8][9][10][11]. The reasons for this interest are many fold: At intermediate and high collision energies partial wave expansion are known to converge very slowly.…”

“…[5] to solve the two-dimensional LS equation with a model two-nucleon potential. It is likely that the two-dimensional wavelets will also prove efficient if used as the expansion basis in SVM.…”

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.…”

“…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].…”

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

“…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.…”