Green's function and LDOS for non-relativistic electron pair

The Coulomb Green's function (GF) for non-relativistic charged particle in field of attractive Coulomb force is extended to describe the interaction of two non-relativistic electrons through repulsive Coulomb forces. Closed-form expressions for the GF, in the absence of electron spins, are derived as one-dimensional integrals. The results are then generalized to include electron spins and account for the Pauli exclusion principle. This leads to a final GF composed of two components, one even and the other odd with respect to exchange particles, with closed-form expressions represented as one-dimensional integrals. The Dyson equations for spin-independent potentials is presented. The local density of states (LDOS) is calculated, which is a combination of contributions from both even and odd GFs. This calculation reveals the dependence of LDOS on inter-electron distance and energy. Separate analysis of the impact of the Pauli exclusion principle is provided. An examination of the pseudo-LDOS, arising from the two-body contribution to the Green's function, is undertaken. Complete suppression of the LDOS at~$r=0$ is ensured by this term, which exhibits a restricted spatial extent. The reasons for the emergence of this pseudo-LDOS are elucidated.