  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.
