AbstractAn exact equivalence of s—dexchange model and the degenerate Hubbard model is shown using the atomic operator representation. The analogy of exact Nagaoka's theorem is proved for a narrowband semiconductor with one carrier and infinite exchange interaction. Using the diagram technique for the Hubbard operators the energy of acoustic and optical magnons is calculated in linear approximation on carrier concentration with allowance for both, kinetic energy and intersite Heisenberg exchange interaction.