An iterative approach for the exact solution of the pairing Hamiltonian

A new iterative algorithm is established for the exact solution of the standard pairing problem, based on the Richardson-Gaudin method using the polynomial approach. It provides efficient and robust solutions for both spherical and deformed systems at a large scale. The key to its success is that the initial guess for the solutions of such a large set of the non-linear equations is provided in a physically meaningful and controllable manner. Moreover, one reduces the large-dimensional problem to a one-dimensional Monte Carlo sampling procedure, which improves the algorithm's efficiency and avoids the non-solutions and numerical instabilities that persist in most existing approaches. We calculated the ground state and low-lying excited states of equally spaced systems at different pairing strengths G. We then applied the model to study the quantum phase transitional Sm isotopes and the actinide nuclei Pu isotopes, where an excellent agreement with experimental data is obtained.