On microscopic theory of spin-S Bose-Einstein condensate in a magnetic field

The Bogoliubov model for weakly interacting Bose gas is extended to Bose-Einstein condensation (BEC) of spin-S atoms in a magnetic field. Equation for the vectorial order parameter valid at temperature $T\to 0$ is derived and its particular solution is found. This solution corresponds to the formation of BEC of atoms with a definite spin projection onto direction of a magnetic field. We study the thermodynamic stability of the found solution and obtain the expressions for low-lying collective modes.