Numerical treatment of long-range Coulomb potential with Berggren bases

TL;DR

This paper evaluates and compares different numerical methods for treating the long-range Coulomb potential within Berggren bases in nuclear physics, introducing a new discretization approach that improves accuracy over existing techniques.

Contribution

It introduces a novel discretization scheme for Coulomb kernels in Berggren bases, enhancing precision in complex-energy nuclear physics calculations.

Findings

The new method outperforms traditional cut and analytical techniques in accuracy.

Discretization schemes effectively handle Coulomb singularities in complex-energy bases.

Application to sd-shell proton states demonstrates practical effectiveness.

Abstract

The Schrodinger equation incorporating the long-range Coulomb potential takes the form of a Fredholm equation whose kernel is singular on its diagonal when represented by a basis bearing a continuum of states, such as in a Fourier-Bessel transform. Several methods have been devised to tackle this difficulty, from simply removing the infinite-range of the Coulomb potential with a screening or cut function to using discretizing schemes which take advantage of the integrable character of Coulomb kernel singularities. However, they have never been tested in the context of Berggren bases, which allow many-body nuclear wave functions to be expanded, with halo or resonant properties within a shell model framework. It is thus the object of this paper to test different discretization schemes of the Coulomb potential kernel in the framework of complex-energy nuclear physics. For that, the Berggren basis expansion of proton states pertaining to the sd-shell arising in the A ~ 20 region, being typically resonant, will be effected. Apart from standard frameworks involving a cut function or analytical integration of singularities, a new method will be presented, which replaces diagonal singularities by finite off-diagonal terms. It will be shown that this methodology surpasses in precision the two former techniques.