Singular Short Range Potentials in the J-Matrix Approach

TL;DR

This paper applies the J-matrix method to evaluate the S-matrix and bound states for singular screened Coulomb potentials, effectively handling singularities analytically and demonstrating accuracy near the bound-unbound transition.

Contribution

It introduces a novel application of the J-matrix approach to singular Coulomb potentials, combining analytical and numerical techniques for improved accuracy.

Findings

Accurately computes bound and resonance energies for singular potentials.

Handles the 1/r singularity analytically within the J-matrix framework.

Shows favorable comparison with existing numerical data.

Abstract

We use the tools of the J-matrix method to evaluate the S-matrix and then deduce the bound and resonance states energies for singular screened Coulomb potentials, both analytic and piecewise differentiable. The J-matrix approach allows us to absorb the 1/r singularity of the potential in the reference Hamiltonian, which is then handled analytically. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian. The remaining part of the potential, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. To exhibit the power of our approach we have considered the most delicate region close to the bound-unbound transition and compared our results favorably with available numerical data.