Hubbard model on d-dimensional hypercubes: Exact solution for the two-electron case

The Hubbard model is exactly solved for two particles with opposite spins on d-dimensional hypercubes. It is shown that the spectrum can be separated into two parts: a trivial (U-independent) part resulting from symmetries of hypercubes and a non-trivial part expressed as a single-impurity problem on a set of finite chains of size $d^\prime+1$ ($d^\prime \le d$). The exact expression for the one-particle Green's function is given. Finally, we discuss the extension of these results to standard hypercubic lattices with periodic boundary conditions.