-t & i,j\mbox{-nearest neighbours} \\ 
0 & \mbox{otherwise}.
\end{array}
\right.$$ Then

$$\varepsilon _{\mathbf{k}}=-2t(\cos k_xa+\cos k_ya+\cos k_za).$$

The hopping parameter $t$ is simply related to the bandwidth $W$ of the Bloch band (2), e. g. $W=12t$ for the sc lattice$.$ The interplay between the two model parameters, $W$ and $U,$ is decisive for the properties of the model resulting in strong electron correlations, leading to band magnetism (see e.g. Refs \[2\], \[3\] for a review), insulator-to-metal transition (see e.g. Refs \[3\], \[4\] and papers cited therein) and high-$T_c$ superconductivity (negative $U$-model, see e.g. Ref. \[5\]). The Hubbard model very often plays also a role of a submodel for many other more complicated models (as e.g. Anderson model (see Ref. \[6\]), s-f model (see Refs \[7,8\]) and so on). Especially interesting, but difficult to handle are the properties of the Hubbard model in such a regime of the model parameters where the bandwidth $W$ is comparable to the Coulomb repulsion $U$. In a large number of papers \[9-30\] the authors tried to solve this model using many sophisticated methods. The exact solution, however, does not exist till now. Many authors tried to change the situation in this field by introducing the expansion 