  The fermionic and bosonic sectors of the 2-site Hubbard model have been
exactly solved by means of the equation of motion and Green's function
formalism. The exact solution of the t-J model has been also reported to
investigate the low-energy dynamics. We have successfully searched for the
exact eigenoperators, and the corresponding eigenenergies, having in mind the
possibility to use them as an operatorial basis on the lattice. Many local,
single-particle, thermodynamical and response properties have been studied as
functions of the external parameters and compared between the two models and
with some numerical and exact results. It has been shown that the 2-site
Hubbard model already contains the most relevant energy scales of the Hubbard
model: the local Coulomb interaction $U$ and the spin-exchange one
$J=\frac{4t^2}U$. As a consequence of this, for some relevant properties
(kinetic energy, double occupancy, energy, specific heat and entropy) and as
regards the metal-insulator transition issue, it has resulted possible to
almost exactly mime the behavior of larger systems, sometimes using a higher
temperature to get a comparable level spacing. The 2-site models have been also
used as toy models to test the efficiency of the Green's function formalism for
composite operators. The capability to reproduce the exact solutions, obtained
by the exact diagonalization technique, gives a firm ground to the approximate
treatments based on this formalism.
