  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.
