Green's operator for Hamiltonians with Coulomb plus polynomial potentials

TL;DR

This paper develops a method to compute the Green's operator for Hamiltonians with Coulomb plus polynomial potentials using a matrix-valued continued fraction approach, enabling energy level calculations for specific potentials.

Contribution

It introduces a novel matrix-valued continued fraction technique to evaluate Green's operators for Coulomb plus polynomial potentials in the Coulomb-Sturmian basis.

Findings

Green's operator expressed as a matrix-valued continued fraction

Calculated energy levels for Coulomb plus linear and quadratic potentials

Demonstrated the method's effectiveness with explicit examples

Abstract

The Hamiltonian of a Coulomb plus polynomial potential on the Coulomb-Sturmian basis has an infinite symmetric band-matrix structure. A band matrix can always be considered as a block-tridiagonal matrix. So, the corresponding Green's operator can be given as a matrix-valued continued fraction. As examples, we calculate the Green's operator for the Coulomb plus linear and quadratic potential problems and determine the energy levels.