The main goal of the project concerned the investigations of the phenomenon of the Bose-Einstein condensation (ВЕС) in a mixt matter-light system formed by the two-dimensional (2D) magnetoexcitons and by the photons accumulated into the microresonator what leads to the formation of the new mixed quasipaticles named as polaritons. The ВЕС of 2D polaritons on the lower branch of the dispersion law in the point of the in-plane wave vector k_||=0 wase investigated. In such conditions new quantum states as, for example, the coherent superposition of the acoustical plasmon magnetoexciton and photon states of the plasmoriton-type appeared. The ВЕС of atoms and molecules in the traps in the presence of the two-photon Raman quantum transitions wase studied taking into account the tunneling of the particles between the potential wells of the traps.

The initial idea concerning the possibility of the ВЕС and of the superfluidity of the ecxcitons in semiconductors with finite life time in conditions of the quasiequilibrium but far from the thermal equilibrium was suggested and intensively investigated during last 50 years by the collaborators of the Institute of Applied Physics. At present time this possibility was experimentally realized in many laboratories of the world in the variant of the ВЕС of the 2D polaritons in microcavities. The theoretical and experimental investigations in this direction becomes an actual branch of the condensed matter physics [1-6]. The methods of the theoretical physics such as the Green's functions method and the Feynman diagrams technique wase used studying the collective elementary excitations.

The coherent macroscopic state was introduced using the method elaborated by Keldysh, Kopaev, Kozlov. The theory of the weakly nonideal Bose-gas and the Bogoliubov u-v transformation of the creation and annihilation operators of the quasiparticles was used.

The Rabi frequencies, which determine the spread and the splitting of the polariton branches was determined. The collective elementary excitations in the mixed system of 2D magnetoexcitons and of the photons confined into the resonator in the state of the ВЕС of the polaritons was performed in the frame of two models. One of them the explicit participation of the optical and acoustical plasmons, of the magnetoexcitons and of the confined photons takes into account. The second one on the model of the weakly nonideal Bose-gas formed by 2D polarotons is based.