Greengard and Rokhlin`s method of solving certain integral equations consists in dividing the integration interval into N sub-intervals, in each sub-interval expanding the solution into Chebyshev polynomials, and subsequently recombining the sub-intervals into one. We present a new improved form of the recombination procedure, developed in a collaboration between the Physics and Mathematics Departments, which is both fast and numerically stable. We give examples for the solution of the Lippmann-Schwinger equation, and evaluate overlap integrals of two such solutions and a potential. Our method is well suited especially if the integrand is very oscillatory, since the truncation errors can be controlled rigorously.
A new spectral type method for solving the one dimensional quantum-mechanical Lippmann-Schwinger integral equation in configuration space is described. The radial interval is divided into partitions, not necessarily of equal length. Two independent local solutions of the integral equation are obtained in each interval via Clenshaw-Curtis quadrature in terms of Chebyshev Polynomials. The local solutions are then combined into a global solution by solving a matrix equation for the coefficients. This matrix is sparse and the equation is easily soluble. The method shows excellent numerical stability, as is demonstrated by several numerical examples.
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