Iterative solution of integral equations on a basis of positive energy Sturmian eigenfunctions

TL;DR

This paper introduces an improved iterative method for solving integral equations using auxiliary Sturmian functions, achieving high accuracy efficiently, demonstrated on scattering problems with potential applications to Schrödinger equations.

Contribution

The paper presents a novel iterative approach that avoids the need for exact Sturmian functions, enhancing convergence speed and accuracy in solving integral equations.

Findings

Achieved 1:10^6 accuracy after 14 iterations

Achieved 1:10^10 accuracy after 20 iterations

Method extends to general kernels and Schrödinger equations

Abstract

Years ago S. Weinberg suggested the "Quasi-Particle" method (Q-P) for iteratively solving an integral equation, based on an expansion in terms of sturmian functions that are eigenfunctions of the integral kernel. An improvement of this method is presented that does not require knowledge of such sturmian functions, but uses simpler auxiliary sturmian functions instead. This improved Q-P method solves the integral equation iterated to second order so as to accelerate the convergence of the iterations. Numerical examples are given for the solution of the Lippmann-Schwinger integral equation for the scattering of a particle from a potential with a repulsive core. An accuracy of 1:10^6 is achieved after 14 iterations, and 1:10^10 after 20 iterations. The calculations are carried out in configuration space for positive energies with an accuracy of 1:10^11 by using a spectral expansion method in terms of Chebyshev polynomials. The method can be extended to general integral kernels, and also to solving a Schr\"odinger equation with Coulomb or non-local potentials.