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For a three-electron system with finite-strength interactions confined to aone-dimensional harmonic trap, we solve the Schroedinger equation analyticallyto obtain the exact solutions, from which we construct explicitly thesimultaneous eigenstates of the energy and total spin for the first time. Thesolutions for the three-electron system allow us to derive analytic expressionsfor the exact one-particle Green's function (GF) for the correspondingtwo-electron system. We calculate the GF in frequency domain to examinesystematically its behavior depending on the electronic interactions. We alsocompare 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 theenergy gap between the KS system and the original system is larger for astronger interaction. We perform numerical examination on the behavior of GFsin real space to demonstrate that the exact and KS GFs can have shapes quitedifferent from each other. Our simple model will help to understand genericcharacteristics of interacting GFs.

journal of physics. condensed matter

One-particle Green's function of interacting two electrons using  analytic solutions for a three-body problem: comparison with exact Kohn--Sham  system