On the Coulomb-Sturmian matrix elements of the Coulomb Green's operator

TL;DR

This paper derives an exact analytic formula for the Coulomb Green's operator matrix elements using Coulomb-Sturmian basis, involving continued fractions and hypergeometric functions, advancing analytical methods in quantum Coulomb problems.

Contribution

It introduces a novel exact formula for Coulomb Green's matrix elements, connecting continued fractions with hypergeometric functions, enhancing analytical approaches in quantum mechanics.

Findings

Derived an explicit formula for Coulomb Green's matrix elements.

Connected continued fractions with hypergeometric functions.

Provided a new analytical tool for Coulomb Hamiltonian calculations.

Abstract

The two-body Coulomb Hamiltonian, when calculated in Coulomb-Sturmian basis, has an infinite symmetric tridiagonal form, also known as Jacobi matrix form. This Jacobi matrix structure involves a continued fraction representation for the inverse of the Green's matrix. The continued fraction can be transformed to a ratio of two $_{2}F_{1}$ hypergeometric functions. From this result we find an exact analytic formula for the matrix elements of the Green's operator of the Coulomb Hamiltonian.