  For a three-electron system with finite-strength interactions confined to a
one-dimensional harmonic trap, we solve the Schroedinger equation analytically
to obtain the exact solutions, from which we construct explicitly the
simultaneous eigenstates of the energy and total spin for the first time. The
solutions for the three-electron system allow us to derive analytic expressions
for the exact one-particle Green's function (GF) for the corresponding
two-electron system. We calculate the GF in frequency domain to examine
systematically its behavior depending on the electronic interactions. We also
compare the pole structure of non-interacting GF using the exact Kohn--Sham
(KS) potential with that of the exact GF to find that the discrepancy of the
energy gap between the KS system and the original system is larger for a
stronger interaction. We perform numerical examination on the behavior of GFs
in real space to demonstrate that the exact and KS GFs can have shapes quite
different from each other. Our simple model will help to understand generic
characteristics of interacting GFs.
